Nested numerical solving problems

Scientific knowledge often takes the form of specific relationships expressed by systems of equations. For example:

• A system of ordinary differential equations connects some state variables \(x\) with some other variables \(\theta\) with equations with the form \(\frac{dx}{dt}= f(x,\theta, t)\).
• An algebraic equation systems says that some variables \(v\) are related so that \(f(x, \theta) = 0\)
• A differential algebraic equation system says that some state variables’ rates of change have the form \(\frac{dx}{dt}= f(x, \theta, t)\) and that they satisfy some algebraic constraints \(f(x, \theta)=0\).

If there is an analytic solution to the equation system, we can just include the solution in our statistical model like any other form of structural knowledge: easy! However, often we want to solve equations that are hard or impossible to solve analytically, but can be solved approximately using numerical methods.

This is tricky in the context of Hamiltonian Monte Carlo for two reasons:

1. Computation: HMC requires many evaluations of the log probability density function and its gradients.

Important

At every evaluation, the sampler needs to solve the embedded equation system and find the gradients of the solution with respect to all model parameters.

2. Extra source of error: how good of an approximation is good enough?

Reading:

• Timonen et al. (2022)
• Stan user guide sections: algebraic equation systems, ODE systems, DAE systems.

Background

Differential Equations

Are equations that relate functions to their derivatives. Some examples of these functions are quantities such as (1) The volume of the liquid in a bioreactor over time \[ \frac{dV}{dt} = F_{in} - F_{out}. \] (2) The temperature of a steel rod with heat source at one end \[ \frac{dT}{dt} = \alpha \frac{d^{2}T}{dx^{2}}. \] (3) The concentration of a substrate in a bioreactor over time (example below). As mentioned previously, the solution to many of these sorts of differential equations does not have an algebraic solution, such as \(T(t) = f(x, t)\).

Ordinary Differential Equations (ODEs)

Arguably, the most simple type of differential equation is what is known as an ordinary differential equation. This means that there is only one independent parameter, typically this will be either time or a dimension. We will only consider ODEs for the remainder of the course.

To gain some intuition about what is going on we will investigate the height of an initially reactor over time with a constant flow rate into the reactor.

Reactor

\[ \frac{dV}{dt} = F_{in} - F_{out} \] \[ \frac{dh}{dt} = \frac{F_{in} - F_{out}}{Area} \] \[ = \frac{0.1 - 0.02 * h}{1} (\frac{m^3}{min}) \]

This ODE is an example which can be manually integrated and has an analytic solution. We shall investigate how the height changes over time and what the steady state height is.

The first question is an example of an initial-value problem. Where we know the initial height (h=0), and we can solve the integral

\[ dh = \int_{t=0}^{t} 0.1 - 0.02*h dt. \] By using integrating factors (Don’t worry about this) we can solve for height \[ h = \frac{0.1}{0.02} + Ce^{-0.02t}. \] Finally, we can solve for the height by substituting what we know: at \(t = 0\), \(h = 0\). Therefore, we arrive at the final equation \[ h = \frac{0.1}{0.02}(1 - e^{-0.02t}). \]

We can answer the question about what its final height would be by solving for the steady-state

\[ \frac{dh}{dt} = 0. \]

Where after rearranging we find the final height equal to 5m, within the dimensions of the reactor

\[ h = \frac{0.1}{0.02} (m). \]

This is just a primer on differential equations. Solving differential equations using numerical solvers usually involves solving the initial-value problem, but rather than solving it analytically, a (hopefully stable) solver will increment the time and state values dependent on the rate equations. This is also an example of a single differential equation, however, we will investigate systems of equations as well.

Example

We have some tubes containing a substrate \(S\) and some biomass \(C\) that we think approximately follow the Monod equation for microbial growth:

\[\begin{align*} \frac{dX}{dt} &= \frac{\mu_{max}\cdot S(t)}{K_{S} + S(t)}\cdot X(t) \\ \frac{dS}{dt} &= -\gamma \cdot \frac{\mu_{max}\cdot S(t)}{K_{s} + S(t)} \cdot X(t) \end{align*}\]

We measured \(X\) and \(S\) at different timepoints in some experiments and we want to try and find out \(\mu_{max}\), \(K_{S}\) and \(\gamma\) for the different strains in the tubes.

You can read more about the Monod equation in Allen and Waclaw (2019).

What we know

\(\mu_{max}, K_S, \gamma, S, X\) are non-negative.

\(S(0)\) and \(X(0)\) vary a little by tube.

\(\mu_{max}, K_S, \gamma\) vary by strain.

Measurement noise is roughly proportional to measured quantity.

Statistical model

We use two regression models to describe the measurements:

\[\begin{align*} y_X &\sim LN(\ln{\hat{X}}, \sigma_{X}) \\ y_S &\sim LN(\ln{\hat{S}}, \sigma_{S}) \end{align*}\]

To capture the variation in parameters by tube and strain we add a hierarchical regression model:

\[\begin{align*} \ln{\mu_{max}} &\sim N(a_{\mu_{max}}, \tau_{\mu_max}) \\ \ln{\gamma} &\sim N(a_{gamma}, \tau_{\gamma}) \\ \ln{\mu_{K_S}} &\sim N(a_{K_S}, \tau_{K_S}) \end{align*}\]

To get a true abundance given some parameters we put an ode in the model:

\[ \hat{X}(t), \hat{S}(t) = \text{solve-monod-equation}(t, X_0, S_0, \mu_max, \gamma, K_S) \]

imports

import itertools

import arviz as az
import cmdstanpy
import pandas as pd
import numpy as np

from matplotlib import pyplot as plt

No entry for terminal type "xterm-color";
using dumb terminal settings.

Specify true parameters

In order to avoid doing too much annoying handling of strings we assume that all the parts of the problem have meaningful 1-indexed integer labels: for example, species 1 is biomass.

This code specifies the dimensions of our problem.

N_strain = 4
N_tube = 16
N_timepoint = 20
duration = 15
strains = [i+1 for i in range(N_strain)]
tubes = [i+1 for i in range(N_tube)]
species = [1, 2]
measurement_timepoint_ixs = [4, 7, 12, 15, 17]
timepoints = pd.Series(
    np.linspace(0.01, duration, N_timepoint),
    name="time",
    index=range(1, N_timepoint+1)
)
SEED = 12345
rng = np.random.default_rng(seed=SEED)

This code defines some true values for the parameters - we will use these to generate fake data.

true_param_values = {
    "a_mu_max": -1.7,
    "a_ks": -1.3,
    "a_gamma": -0.6,
    "t_mu_max": 0.2,
    "t_ks": 0.3,
    "t_gamma": 0.13,
    "species_zero": [
        [
            np.exp(np.random.normal(-2.1, 0.05)), 
            np.exp(np.random.normal(0.2, 0.05))
        ] for _ in range(N_tube)
    ],
    "sigma_y": [0.08, 0.1],
    "ln_mu_max_z": np.random.normal(0, 1, size=N_strain).tolist(),
    "ln_ks_z": np.random.normal(0, 1, size=N_strain).tolist(),
    "ln_gamma_z": np.random.normal(0, 1, size=N_strain).tolist(),
}
for var in ["mu_max", "ks", "gamma"]:
    true_param_values[var] = np.exp(
        true_param_values[f"a_{var}"]
        + true_param_values[f"t_{var}"] * np.array(true_param_values[f"ln_{var}_z"])
    ).tolist()

A bit of data transformation

This code does some handy transformations on the data using pandas, giving us a table of information about the measurements.

tube_to_strain = pd.Series(
    [
        (i % N_strain) + 1 for i in range(N_tube)  # % operator finds remainder
    ], index=tubes, name="strain"
)
measurements = (
    pd.DataFrame(
        itertools.product(tubes, measurement_timepoint_ixs, species),
        columns=["tube", "timepoint", "species"],
        index=range(1, len(tubes) * len(measurement_timepoint_ixs) * len(species) + 1)
    )
    .join(tube_to_strain, on="tube")
    .join(timepoints, on="timepoint")
)

Generating a Stan input dictionary

This code puts the data in the correct format for cmdstanpy.

stan_input_structure = {
    "N_measurement": len(measurements),
    "N_timepoint": N_timepoint,
    "N_tube": N_tube,
    "N_strain": N_strain,
    "tube": measurements["tube"].values.tolist(),
    "measurement_timepoint": measurements["timepoint"].values.tolist(),
    "measured_species": measurements["species"].values.tolist(),
    "strain": tube_to_strain.values.tolist(),
    "timepoint_time": timepoints.values.tolist(),
}

This code defines some prior distributions for the model’s parameters

priors = {
    # parameters that can be negative:
    "prior_a_mu_max": [-1.8, 0.2],
    "prior_a_ks": [-1.3, 0.1],
    "prior_a_gamma": [-0.5, 0.1],
    # parameters that are non-negative:
    "prior_t_mu_max": [-1.4, 0.1],
    "prior_t_ks": [-1.2, 0.1],
    "prior_t_gamma": [-2, 0.1],
    "prior_species_zero": [[[-2.1, 0.1], [0.2, 0.1]]] * N_tube,
    "prior_sigma_y": [[-2.3, 0.15], [-2.3, 0.15]],
}

The next bit of code lets us configure Stan’s interface to the Sundials ODE solver.

ode_solver_configuration = {
    "abs_tol": 1e-7,
    "rel_tol": 1e-7,
    "max_num_steps": int(1e7)
}

Now we can put all the inputs together

stan_input_common = stan_input_structure | priors | ode_solver_configuration

Load the model

This code loads the Stan program at monod.stan as a CmdStanModel object and compiles it using cmdstan’s compiler.

model = cmdstanpy.CmdStanModel(stan_file="../src/stan/monod.stan")
print(model.code())

functions {
  real get_mu_at_t(real mu_max, real ks, real S_at_t) {
    return (mu_max * S_at_t) / (ks + S_at_t);
  }
  vector ddt(real t, vector species, real mu_max, real ks, real gamma) {
    real mu_at_t = get_mu_at_t(mu_max, ks, species[2]);
    vector[2] out;
    out[1] = mu_at_t * species[1];
    out[2] = -gamma * mu_at_t * species[1];
    return out;
  }
}
data {
  int<lower=1> N_measurement;
  int<lower=1> N_timepoint;
  int<lower=1> N_tube;
  int<lower=1> N_strain;
  array[N_measurement] int<lower=1, upper=N_tube> tube;
  array[N_measurement] int<lower=1, upper=N_timepoint> measurement_timepoint;
  array[N_measurement] int<lower=1, upper=2> measured_species;
  vector<lower=0>[N_measurement] y;
  array[N_tube] int<lower=1, upper=N_strain> strain;
  array[N_timepoint] real<lower=0> timepoint_time;
  array[N_tube, 2] vector[2] prior_species_zero;
  array[2] vector[2] prior_sigma_y;
  vector[2] prior_a_mu_max;
  vector[2] prior_a_ks;
  vector[2] prior_a_gamma;
  vector[2] prior_t_mu_max;
  vector[2] prior_t_gamma;
  vector[2] prior_t_ks;
  real<lower=0> abs_tol;
  real<lower=0> rel_tol;
  int<lower=1> max_num_steps;
  int<lower=0, upper=1> likelihood;
}
parameters {
  vector[N_strain] ln_mu_max_z;
  vector[N_strain] ln_ks_z;
  vector[N_strain] ln_gamma_z;
  real a_mu_max;
  real a_ks;
  real a_gamma;
  real<lower=0> t_mu_max;
  real<lower=0> t_ks;
  real<lower=0> t_gamma;
  array[N_tube] vector<lower=0>[2] species_zero;
  vector<lower=0>[2] sigma_y;
}
transformed parameters {
  vector[N_strain] mu_max = exp(a_mu_max + ln_mu_max_z * t_mu_max);
  vector[N_strain] ks = exp(a_ks + ln_ks_z * t_ks);
  vector[N_strain] gamma = exp(a_gamma + ln_gamma_z * t_gamma);
  array[N_tube, N_timepoint] vector[2] abundance;
  for (tube_t in 1 : N_tube) {
    abundance[tube_t] = ode_bdf_tol(ddt, species_zero[tube_t], 0,
                                    timepoint_time,
                                    abs_tol, rel_tol, max_num_steps,
                                    mu_max[strain[tube_t]],
                                    ks[strain[tube_t]], gamma[strain[tube_t]]);
  }
}
model {
  // priors
  ln_mu_max_z ~ std_normal();
  ln_ks_z ~ std_normal();
  ln_gamma_z ~ std_normal();
  a_mu_max ~ normal(prior_a_mu_max[1], prior_a_mu_max[2]);
  a_ks ~ normal(prior_a_ks[1], prior_a_ks[2]);
  a_gamma ~ normal(prior_a_gamma[1], prior_a_gamma[2]);
  t_mu_max ~ lognormal(prior_t_mu_max[1], prior_t_mu_max[2]);
  t_ks ~ lognormal(prior_t_ks[1], prior_t_ks[2]);
  t_gamma ~ lognormal(prior_t_gamma[1], prior_t_gamma[2]);
  for (s in 1 : 2) {
    sigma_y[s] ~ lognormal(prior_sigma_y[s, 1], prior_sigma_y[s, 2]);
    for (t in 1 : N_tube){
      species_zero[t, s] ~ lognormal(prior_species_zero[t, s, 1],
                                     prior_species_zero[t, s, 2]);
    }
  }
  // likelihood
  if (likelihood) {
    for (m in 1 : N_measurement) {
      real yhat = abundance[tube[m], measurement_timepoint[m], measured_species[m]];
      y[m] ~ lognormal(log(yhat), sigma_y[measured_species[m]]);
    }
  }
}
generated quantities {
  vector[N_measurement] yrep;
  vector[N_measurement] llik;
  for (m in 1 : N_measurement){
    real yhat = abundance[tube[m], measurement_timepoint[m], measured_species[m]];
    yrep[m] = lognormal_rng(log(yhat), sigma_y[measured_species[m]]);
    llik[m] = lognormal_lpdf(y[m] | log(yhat), sigma_y[measured_species[m]]);
  }
}

Sample in fixed param mode to generate fake data

stan_input_true = stan_input_common | {
    "y": np.ones(len(measurements)).tolist(),  # dummy values as we don't need measurements yet
    "likelihood": 0                            # we don't need to evaluate the likelihood
}
coords = {
    "strain": strains,
    "tube": tubes,
    "species": species,
    "timepoint": timepoints.index.values,
    "measurement": measurements.index.values
}
dims = {
    "abundance": ["tube", "timepoint", "species"],
    "mu_max": ["strain"],
    "ks": ["strain"],
    "gamma": ["strain"],
    "species_zero": ["tube", "species"],
    "y": ["measurement"],
    "yrep": ["measurement"],
    "llik": ["measurement"]
}

mcmc_true = model.sample(
    data=stan_input_true,
    iter_sampling=1,
    fixed_param=True,
    chains=1,
    refresh=1,
    inits=true_param_values,
    seed=SEED,
)
idata_true = az.from_cmdstanpy(
    mcmc_true,
    dims=dims,
    coords=coords,
    posterior_predictive={"y": "yrep"},
    log_likelihood="llik"
)

09:25:29 - cmdstanpy - INFO - CmdStan start processing
09:25:29 - cmdstanpy - INFO - CmdStan done processing.

Look at results

def plot_sim(true_abundance, fake_measurements, species_to_ax):
    f, axes = plt.subplots(1, 2, figsize=[9, 3])

    axes[species_to_ax[1]].set_title("Species 1")
    axes[species_to_ax[2]].set_title("Species 2")
    for ax in axes:
        ax.set_xlabel("Time")
        ax.set_ylabel("Abundance")
        for (tube_i, species_i), df_i in true_abundance.groupby(["tube", "species"]):
            ax = axes[species_to_ax[species_i]]
            fm = df_i.merge(
                fake_measurements.drop("time", axis=1),
                on=["tube", "species", "timepoint"]
            )
            ax.plot(
                df_i.set_index("time")["abundance"], color="black", linewidth=0.5
            )
            ax.scatter(
                fm["time"],
                fm["simulated_measurement"],
                color="r",
                marker="x",
                label="simulated measurement"
            )
    return f, axes

species_to_ax = {1: 0, 2: 1}
true_abundance = (
    idata_true.posterior["abundance"]
    .to_dataframe()
    .droplevel(["chain", "draw"])
    .join(timepoints, on="timepoint")
    .reset_index()
)
fake_measurements = measurements.join(
    idata_true.posterior_predictive["yrep"]
    .to_series()
    .droplevel(["chain", "draw"])
    .rename("simulated_measurement")
).copy()
f, axes = plot_sim(true_abundance, fake_measurements, species_to_ax)

f.savefig("img/monod_simulated_data.png")

Sample in prior mode

stan_input_prior = stan_input_common | {
    "y": fake_measurements["simulated_measurement"],
    "likelihood": 0
}
mcmc_prior = model.sample(
    data=stan_input_prior,
    iter_warmup=100,
    iter_sampling=100,
    chains=1,
    refresh=1,
    save_warmup=True,
    inits=true_param_values,
    seed=SEED,
)
idata_prior = az.from_cmdstanpy(
    mcmc_prior,
    dims=dims,
    coords=coords,
    posterior_predictive={"y": "yrep"},
    log_likelihood="llik"
)
idata_prior

09:25:29 - cmdstanpy - INFO - CmdStan start processing
09:25:50 - cmdstanpy - INFO - CmdStan done processing.
09:25:50 - cmdstanpy - WARNING - Non-fatal error during sampling:
Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: initial state[2] is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
Consider re-running with show_console=True if the above output is unclear!

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 564kB
  Dimensions:            (chain: 1, draw: 100, ln_mu_max_z_dim_0: 4,
                          ln_ks_z_dim_0: 4, ln_gamma_z_dim_0: 4, tube: 16,
                          species: 2, sigma_y_dim_0: 2, strain: 4, timepoint: 20)
  Coordinates:
    * chain              (chain) int64 8B 0
    * draw               (draw) int64 800B 0 1 2 3 4 5 6 ... 93 94 95 96 97 98 99
    * ln_mu_max_z_dim_0  (ln_mu_max_z_dim_0) int64 32B 0 1 2 3
    * ln_ks_z_dim_0      (ln_ks_z_dim_0) int64 32B 0 1 2 3
    * ln_gamma_z_dim_0   (ln_gamma_z_dim_0) int64 32B 0 1 2 3
    * tube               (tube) int64 128B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    * species            (species) int64 16B 1 2
    * sigma_y_dim_0      (sigma_y_dim_0) int64 16B 0 1
    * strain             (strain) int64 32B 1 2 3 4
    * timepoint          (timepoint) int64 160B 1 2 3 4 5 6 ... 15 16 17 18 19 20
  Data variables: (12/15)
      ln_mu_max_z        (chain, draw, ln_mu_max_z_dim_0) float64 3kB -1.257 .....
      ln_ks_z            (chain, draw, ln_ks_z_dim_0) float64 3kB 1.134 ... 0.0...
      ln_gamma_z         (chain, draw, ln_gamma_z_dim_0) float64 3kB 2.278 ... ...
      a_mu_max           (chain, draw) float64 800B -1.757 -1.779 ... -1.9 -1.544
      a_ks               (chain, draw) float64 800B -1.382 -1.365 ... -1.373
      a_gamma            (chain, draw) float64 800B -0.5929 -0.6364 ... -0.4605
      ...                 ...
      species_zero       (chain, draw, tube, species) float64 26kB 0.1343 ... 1...
      sigma_y            (chain, draw, sigma_y_dim_0) float64 2kB 0.103 ... 0.1167
      mu_max             (chain, draw, strain) float64 3kB 0.1261 0.179 ... 0.1866
      ks                 (chain, draw, strain) float64 3kB 0.3584 ... 0.2555
      gamma              (chain, draw, strain) float64 3kB 0.7393 ... 0.6419
      abundance          (chain, draw, tube, timepoint, species) float64 512kB ...
  Attributes:
      created_at:                 2024-04-30T07:25:50.742269
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 100
    • ln_mu_max_z_dim_0: 4
    • ln_ks_z_dim_0: 4
    • ln_gamma_z_dim_0: 4
    • tube: 16
    • species: 2
    • sigma_y_dim_0: 2
    • strain: 4
    • timepoint: 20
  • Coordinates: (10)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 6 ... 94 95 96 97 98 99

      array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99])

    • ln_mu_max_z_dim_0
      (ln_mu_max_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_ks_z_dim_0
      (ln_ks_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_gamma_z_dim_0
      (ln_gamma_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • tube
      (tube)
      int64
      1 2 3 4 5 6 7 ... 11 12 13 14 15 16

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16])

    • species
      (species)
      int64
      1 2

      array([1, 2])

    • sigma_y_dim_0
      (sigma_y_dim_0)
      int64
      0 1

      array([0, 1])

    • strain
      (strain)
      int64
      1 2 3 4

      array([1, 2, 3, 4])

    • timepoint
      (timepoint)
      int64
      1 2 3 4 5 6 7 ... 15 16 17 18 19 20

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
             19, 20])

  • Data variables: (15)
    • ln_mu_max_z
      (chain, draw, ln_mu_max_z_dim_0)
      float64
      -1.257 0.1461 ... 1.044 -0.5285

      array([[[-1.25704e+00,  1.46123e-01,  2.73110e-01,  4.32168e-01],
              [ 2.56057e-01,  8.82222e-01,  4.86352e-01,  4.37784e-01],
              [ 1.56385e-01,  6.41261e-01,  8.94836e-01, -8.74206e-01],
              [-5.26406e-01,  8.32781e-01,  1.04478e+00, -8.83506e-01],
              [-1.18894e+00,  4.13661e-01,  1.65888e+00, -8.00048e-01],
              [-9.56987e-01,  1.72047e-01,  1.67022e+00, -5.39850e-01],
              [-1.27985e+00,  5.30518e-01,  1.66929e+00, -3.59238e-01],
              [ 4.88124e-01,  1.23319e-01, -8.75954e-01, -4.75639e-01],
              [ 1.37647e+00,  5.60680e-01, -1.89922e-01, -4.67899e-01],
              [-9.41438e-01,  5.40375e-01,  9.70694e-01,  1.48568e+00],
              [-8.28299e-01,  7.29127e-01,  1.21539e+00,  1.51449e+00],
              [ 1.62069e+00, -2.33020e+00, -4.27831e-01,  2.39344e-01],
              [ 9.25246e-01,  3.75749e+00,  1.00298e+00,  2.40226e-01],
              [-9.55549e-01,  4.38197e-01, -8.87448e-01, -1.41593e+00],
              [-3.14705e-01,  9.72369e-02, -4.71582e-01, -5.39760e-01],
              [ 5.88586e-01, -2.64191e-02,  4.45607e-01,  2.95901e-01],
              [-3.92727e-01,  3.70243e-01,  1.61749e-01,  6.04153e-01],
              [-1.93547e-02,  3.36662e-02,  1.61821e+00, -7.94756e-01],
              [ 2.04512e+00, -1.01242e+00, -8.77405e-01,  4.95403e-02],
              [-2.23577e+00,  8.00444e-01,  8.35704e-01, -3.51333e-01],
      ...
              [-3.46576e-01, -9.87028e-01,  6.99222e-02, -6.96743e-01],
              [-4.14577e-01, -9.47469e-01,  3.49767e-01, -9.48599e-01],
              [-3.23495e-01, -8.47136e-01,  7.44553e-02, -9.82510e-01],
              [-2.25596e-01, -1.03296e+00,  1.52149e-01, -5.47013e-01],
              [ 1.56284e-01, -8.12364e-01,  1.94707e-01, -5.81661e-01],
              [ 6.81679e-02, -9.38737e-01,  2.02546e-01, -3.17898e-01],
              [ 7.69469e-01,  1.41813e+00,  2.54130e-01, -1.13343e-01],
              [ 1.65735e-01, -2.36288e+00,  7.94466e-01, -2.85698e-01],
              [ 6.03559e-03, -4.06089e-01,  3.70685e-01,  1.65174e+00],
              [ 6.04777e-01, -8.71298e-01,  6.13415e-01,  1.81069e+00],
              [ 2.84444e-01,  1.36950e+00, -5.64750e-01, -9.38357e-01],
              [ 2.61676e-01,  1.31203e+00, -3.31694e-01, -8.71329e-01],
              [ 2.75732e-01,  1.43518e+00, -5.99524e-01, -3.38711e-01],
              [ 5.45669e-01,  1.78253e+00, -3.90877e-02, -3.59052e-01],
              [-5.39393e-01, -2.20093e+00,  4.80162e-02,  3.88923e-01],
              [ 5.81911e-01,  2.11277e+00, -1.54841e-01, -4.09968e-01],
              [-5.10658e-01,  7.42973e-01,  7.60975e-01, -1.75922e+00],
              [-3.19379e-01,  7.05254e-01,  1.71299e-02, -9.75154e-01],
              [-2.01322e-01,  1.01129e+00,  6.39685e-02, -1.37137e+00],
              [-1.46123e-01,  3.43615e-01,  1.04388e+00, -5.28545e-01]]])

    • ln_ks_z
      (chain, draw, ln_ks_z_dim_0)
      float64
      1.134 -0.829 ... 0.2943 0.02748

      array([[[ 1.13447e+00, -8.29042e-01,  1.11660e+00,  9.67294e-01],
              [-1.28993e+00,  1.32964e+00, -1.66244e+00, -1.47922e+00],
              [-8.69819e-01, -1.04942e+00, -6.38453e-01, -4.93959e-01],
              [-1.47279e+00, -7.87440e-01, -4.10423e-02, -9.45293e-01],
              [-1.50744e+00, -4.08436e-01,  1.38305e-02, -7.15677e-01],
              [-1.19211e+00, -6.57574e-01,  3.87019e-02, -6.49278e-01],
              [-1.36825e+00, -4.68140e-01,  3.83407e-01, -4.65166e-01],
              [-2.04824e-01, -6.38397e-01, -3.65344e-01, -3.56069e-01],
              [ 4.47184e-01, -1.82079e+00, -3.76054e-01, -1.22784e+00],
              [-1.32853e-02, -5.49132e-01,  1.26031e+00,  1.07134e+00],
              [ 9.98497e-02, -7.70141e-01,  1.04351e+00,  1.16496e+00],
              [-3.42033e-01, -2.73159e-01, -7.74233e-01, -1.02210e+00],
              [ 1.07393e+00, -7.23112e-02,  7.22464e-01, -9.71724e-01],
              [ 4.79573e-01,  1.54350e+00,  8.68416e-01, -1.53499e+00],
              [ 2.68485e-01,  1.30821e+00,  5.29005e-01, -1.84095e+00],
              [-3.36788e-01, -1.37296e+00, -9.48244e-01,  2.07064e+00],
              [-1.09735e-01,  1.21872e+00,  1.53517e+00, -2.52577e+00],
              [ 1.59823e+00,  7.04790e-01, -1.79301e+00, -5.79087e-01],
              [-1.82652e-01, -9.60484e-01, -2.21627e-01,  1.56755e-01],
              [-1.08297e+00,  2.18051e+00,  1.75813e+00,  3.31010e-01],
      ...
              [-1.54857e+00,  4.58905e-02,  4.61585e-01,  3.13846e-02],
              [-1.30941e+00,  2.90785e-01,  6.70938e-01,  2.27883e-01],
              [-1.48024e+00,  2.44321e-01,  7.64641e-01,  2.89488e-01],
              [-1.62321e+00,  4.68740e-01,  6.90326e-01,  5.88403e-01],
              [-1.55745e+00,  7.03937e-01,  7.84584e-01,  7.90465e-01],
              [-1.61507e+00,  8.46970e-01,  5.68398e-01,  8.82636e-01],
              [ 1.95775e+00, -1.29133e+00, -2.89391e-01, -7.22797e-01],
              [-1.11505e+00,  1.71287e+00, -4.14685e-01,  7.78635e-01],
              [ 7.39809e-01,  1.49739e+00, -7.15864e-01, -3.53607e-01],
              [ 5.25094e-01,  1.59555e+00, -1.39372e+00, -8.73427e-01],
              [-9.93423e-01, -9.09712e-01,  9.76356e-01,  1.60998e+00],
              [-8.98714e-01, -8.94382e-01,  1.00001e+00,  1.55464e+00],
              [-1.12771e+00, -1.58252e+00,  1.10339e+00,  1.82835e+00],
              [-8.42594e-01, -1.41372e+00,  1.01521e+00,  1.21156e+00],
              [ 8.08869e-01,  1.24786e+00, -9.89164e-01, -1.40039e+00],
              [-8.01547e-01, -1.22294e+00,  9.63238e-01,  1.45322e+00],
              [ 1.16044e-01,  1.86354e+00,  5.85893e-01, -9.06682e-02],
              [ 4.36406e-01,  1.53318e+00,  2.58617e-01, -7.70810e-02],
              [ 2.58445e-01,  1.27813e+00,  1.00048e-01,  6.49979e-05],
              [-2.37000e-01,  9.17239e-01,  2.94326e-01,  2.74846e-02]]])

    • ln_gamma_z
      (chain, draw, ln_gamma_z_dim_0)
      float64
      2.278 -0.02889 ... -1.083 0.1159

      array([[[ 2.27760e+00, -2.88877e-02,  8.76825e-01,  1.66738e+00],
              [-1.64364e+00,  4.74637e-01, -5.73694e-01, -1.19532e+00],
              [-1.05329e+00, -5.25101e-02, -1.67713e-01,  1.70110e-02],
              [-6.55002e-02,  1.71417e-01, -8.87581e-01, -1.52869e-01],
              [-1.16346e-01,  3.70469e-01, -8.94486e-01, -2.13148e-03],
              [-3.88650e-01,  2.33648e-01, -1.08723e+00,  1.62565e-02],
              [-2.00740e-01,  1.28711e-01, -6.52958e-01, -1.79686e-01],
              [-1.01575e+00,  9.28409e-01,  6.64145e-01, -7.43491e-01],
              [-5.57048e-01,  1.05764e+00,  5.93314e-01, -9.09131e-01],
              [ 3.62974e-01, -2.09723e+00, -1.67337e+00,  4.73235e-01],
              [ 1.37006e-01, -2.36636e+00, -1.73586e+00,  6.64678e-01],
              [-5.44031e-01,  8.03165e-01, -1.08477e+00, -6.02024e-01],
              [ 9.18607e-01,  5.67952e-01,  6.45011e-01,  1.61773e+00],
              [ 5.65692e-02, -2.01694e-01,  2.21249e-01,  2.73169e+00],
              [ 4.06184e-01, -1.39372e-01,  6.75807e-01,  1.94060e+00],
              [-6.74575e-01,  4.76567e-01, -8.19501e-01, -1.74667e+00],
              [ 1.25737e+00, -5.03754e-01,  6.01525e-01,  1.88017e+00],
              [-7.10542e-01, -9.78554e-01,  3.39476e+00, -5.65801e-01],
              [ 7.35581e-01, -1.38395e+00, -1.69451e+00, -7.19961e-01],
              [ 1.00276e+00,  2.40158e-01, -3.32907e-01, -5.84841e-01],
      ...
              [ 7.29084e-01,  1.49495e+00, -2.41163e-01, -5.02042e-01],
              [ 5.42134e-01,  1.24865e+00, -1.29268e-01, -6.51325e-01],
              [ 5.55109e-01,  1.07846e+00, -4.12195e-01, -9.50034e-01],
              [ 6.88304e-01,  1.24574e+00, -6.01789e-01, -1.33979e+00],
              [ 5.50412e-01,  1.14586e+00, -7.30567e-01, -1.10855e+00],
              [ 5.24924e-01,  1.34795e+00, -8.17920e-01, -9.33201e-01],
              [ 1.23902e-01, -8.14864e-01,  5.73327e-01,  4.58800e-01],
              [-6.15506e-01,  9.92127e-01, -6.96236e-01, -3.23639e-01],
              [-1.40420e+00, -3.47010e-01,  1.72192e+00,  9.82907e-02],
              [-1.27051e+00, -1.21623e-01,  1.42244e+00,  9.49882e-02],
              [-4.72647e-02,  3.99600e-02, -1.06251e+00, -1.07430e+00],
              [-6.35020e-02,  1.06957e-01, -1.09373e+00, -1.17777e+00],
              [ 8.86054e-02, -1.13070e-01, -1.55408e+00, -2.05000e+00],
              [ 5.80039e-01,  1.38333e-01, -1.71378e+00, -1.71664e+00],
              [-2.70730e-01,  1.58018e-01,  1.71495e+00,  1.90923e+00],
              [ 3.09839e-01, -2.03700e-01, -1.67030e+00, -1.96502e+00],
              [ 1.39508e-01,  1.23081e+00,  7.61668e-01, -7.28978e-02],
              [ 7.37014e-01,  1.28111e+00,  4.66361e-01, -3.73307e-01],
              [ 8.53226e-01,  1.36057e+00,  2.81113e-01, -1.23072e-01],
              [ 9.08523e-01,  1.04120e+00, -1.08306e+00,  1.15864e-01]]])

    • a_mu_max
      (chain, draw)
      float64
      -1.757 -1.779 ... -1.9 -1.544

      array([[-1.75664, -1.7787 , -1.84261, -1.84048, -1.82423, -1.85014,
              -1.88075, -1.51907, -2.07709, -1.9377 , -1.6609 , -1.79077,
              -1.44929, -1.46443, -1.70372, -1.60286, -1.59086, -1.57619,
              -1.86153, -1.73362, -1.66933, -1.681  , -1.79995, -1.80536,
              -1.80386, -1.8567 , -1.97247, -2.03835, -1.96326, -2.14831,
              -1.89239, -1.89815, -1.84648, -1.66977, -1.97688, -1.96273,
              -1.65747, -1.78824, -1.74885, -1.84379, -1.81665, -1.65075,
              -2.14301, -2.02827, -1.70237, -1.87795, -1.52438, -1.58486,
              -1.42518, -1.72836, -1.72474, -1.69481, -1.66279, -1.98461,
              -1.9138 , -1.64293, -1.69642, -1.90439, -2.02058, -1.60217,
              -1.72861, -1.87815, -1.83271, -2.02774, -1.99079, -2.09942,
              -1.99485, -1.91802, -2.19562, -2.24523, -1.93956, -2.07604,
              -1.40656, -1.4896 , -1.67486, -1.47046, -1.71897, -1.86369,
              -1.76828, -1.60361, -1.47272, -2.09033, -2.01939, -1.81867,
              -2.00582, -1.98071, -1.79801, -1.89882, -1.85197, -1.89697,
              -2.02278, -1.99713, -1.28569, -2.06819, -1.84366, -1.78694,
              -1.81975, -2.00338, -1.89973, -1.54356]])

    • a_ks
      (chain, draw)
      float64
      -1.382 -1.365 ... -1.273 -1.373

      array([[-1.38212 , -1.3655  , -1.13153 , -1.17459 , -1.38064 , -1.28488 ,
              -1.32014 , -1.27536 , -1.2699  , -1.35835 , -1.38595 , -1.40225 ,
              -1.19755 , -1.2894  , -1.35986 , -1.45957 , -1.27646 , -1.23201 ,
              -1.1629  , -1.16622 , -1.17444 , -1.16062 , -1.38351 , -1.46336 ,
              -1.46196 , -1.13694 , -1.30183 , -1.3493  , -1.40456 , -1.13336 ,
              -1.4043  , -1.38161 , -1.21283 , -1.42875 , -1.31663 , -1.33839 ,
              -1.36959 , -1.19349 , -1.31183 , -1.33941 , -1.40941 , -1.25555 ,
              -1.13697 , -1.15534 , -1.433   , -1.16668 , -1.20874 , -1.4214  ,
              -1.3273  , -1.30638 , -1.34271 , -1.38923 , -1.41493 , -1.6761  ,
              -1.36621 , -1.29036 , -1.29901 , -1.27018 , -1.31727 , -1.33905 ,
              -1.34421 , -1.31401 , -1.17771 , -1.40293 , -1.17506 , -1.09423 ,
              -1.47736 , -1.45857 , -1.14901 , -1.45347 , -1.06967 , -1.53195 ,
              -1.50125 , -1.38914 , -1.20925 , -1.36351 , -1.25026 , -1.15031 ,
              -1.19809 , -1.10814 , -1.49676 , -0.972995, -1.56917 , -1.15727 ,
              -1.42895 , -1.29807 , -1.15632 , -1.15412 , -1.50157 , -1.51296 ,
              -1.42    , -1.41419 , -1.13851 , -1.35652 , -1.1406  , -1.13623 ,
              -1.18571 , -1.24463 , -1.27345 , -1.37344 ]])

    • a_gamma
      (chain, draw)
      float64
      -0.5929 -0.6364 ... -0.5794 -0.4605

      array([[-0.592897, -0.636445, -0.530916, -0.548767, -0.41845 , -0.510603,
              -0.482938, -0.501066, -0.523522, -0.524803, -0.520775, -0.361777,
              -0.661413, -0.611701, -0.502222, -0.363535, -0.518314, -0.548804,
              -0.530696, -0.503341, -0.381453, -0.712704, -0.352612, -0.40736 ,
              -0.445765, -0.554297, -0.62494 , -0.394142, -0.382285, -0.606243,
              -0.273004, -0.476237, -0.452628, -0.431812, -0.809057, -0.736356,
              -0.320153, -0.625831, -0.508725, -0.636462, -0.51195 , -0.454108,
              -0.344288, -0.312116, -0.519933, -0.482481, -0.675034, -0.657394,
              -0.637295, -0.677753, -0.698967, -0.385939, -0.371044, -0.445633,
              -0.680363, -0.30382 , -0.692481, -0.389836, -0.561865, -0.542851,
              -0.560209, -0.55142 , -0.383798, -0.440125, -0.541291, -0.466337,
              -0.525984, -0.695299, -0.308861, -0.689351, -0.302832, -0.598135,
              -0.561906, -0.508955, -0.491443, -0.504463, -0.5236  , -0.51573 ,
              -0.457735, -0.444903, -0.536438, -0.492495, -0.551661, -0.433587,
              -0.728966, -0.48227 , -0.548365, -0.533625, -0.527194, -0.478241,
              -0.423171, -0.629413, -0.459531, -0.552647, -0.393814, -0.394326,
              -0.440149, -0.379266, -0.579361, -0.460463]])

    • t_mu_max
      (chain, draw)
      float64
      0.25 0.2513 0.2433 ... 0.2305 0.256

      array([[0.250027, 0.251268, 0.243294, 0.275054, 0.208164, 0.252327,
              0.240801, 0.242784, 0.246601, 0.212007, 0.259461, 0.244903,
              0.257267, 0.247343, 0.264167, 0.247671, 0.263706, 0.231281,
              0.202615, 0.210445, 0.222926, 0.22771 , 0.253634, 0.255338,
              0.252656, 0.239722, 0.229544, 0.285141, 0.297053, 0.231065,
              0.245871, 0.249983, 0.226369, 0.201312, 0.301779, 0.291477,
              0.23627 , 0.279103, 0.264347, 0.275687, 0.273781, 0.216982,
              0.302681, 0.286689, 0.239825, 0.252847, 0.238478, 0.231182,
              0.243853, 0.258878, 0.261089, 0.250533, 0.270146, 0.233927,
              0.277487, 0.278498, 0.217784, 0.256461, 0.244892, 0.244619,
              0.253658, 0.248164, 0.231708, 0.32951 , 0.240288, 0.253168,
              0.247043, 0.227376, 0.275881, 0.220109, 0.278331, 0.222786,
              0.220843, 0.25201 , 0.241599, 0.250626, 0.233342, 0.253909,
              0.259399, 0.291512, 0.209401, 0.25419 , 0.24208 , 0.278665,
              0.295326, 0.216797, 0.242381, 0.236982, 0.276927, 0.242949,
              0.247759, 0.275206, 0.236281, 0.208451, 0.263316, 0.264472,
              0.26121 , 0.269693, 0.230503, 0.256049]])

    • t_ks
      (chain, draw)
      float64
      0.3138 0.3274 ... 0.2977 0.3256

      array([[0.3138  , 0.327373, 0.357032, 0.332894, 0.323699, 0.283258,
              0.320167, 0.288913, 0.282575, 0.326193, 0.262087, 0.324408,
              0.265275, 0.253451, 0.280023, 0.297532, 0.343448, 0.247917,
              0.236759, 0.239591, 0.236458, 0.300373, 0.303531, 0.322758,
              0.34428 , 0.264271, 0.322225, 0.296557, 0.298837, 0.344094,
              0.305228, 0.331759, 0.295843, 0.323893, 0.263759, 0.282076,
              0.305759, 0.303103, 0.269853, 0.25359 , 0.270955, 0.328698,
              0.279061, 0.284569, 0.342472, 0.264655, 0.284641, 0.261504,
              0.269906, 0.280504, 0.283948, 0.294667, 0.309748, 0.262143,
              0.233746, 0.328234, 0.273775, 0.334126, 0.277908, 0.270337,
              0.266277, 0.308953, 0.319406, 0.25543 , 0.241048, 0.294196,
              0.28776 , 0.309636, 0.297537, 0.306219, 0.338113, 0.27526 ,
              0.261826, 0.321698, 0.281452, 0.310968, 0.282975, 0.240941,
              0.264812, 0.256692, 0.357123, 0.247116, 0.37448 , 0.276792,
              0.319254, 0.291425, 0.274901, 0.27499 , 0.281966, 0.259697,
              0.238668, 0.262624, 0.326665, 0.30962 , 0.297234, 0.294117,
              0.306963, 0.32622 , 0.297658, 0.325618]])

    • t_gamma
      (chain, draw)
      float64
      0.1277 0.1338 ... 0.1359 0.1482

      array([[0.127728, 0.133758, 0.126569, 0.126135, 0.146703, 0.120594,
              0.145815, 0.137649, 0.136116, 0.132811, 0.159526, 0.144618,
              0.13104 , 0.135085, 0.132214, 0.132927, 0.147185, 0.127779,
              0.113953, 0.12166 , 0.14897 , 0.144162, 0.149865, 0.141849,
              0.135691, 0.134132, 0.130698, 0.131609, 0.114727, 0.150347,
              0.116683, 0.130158, 0.150532, 0.128748, 0.122394, 0.112366,
              0.162417, 0.113395, 0.136201, 0.135861, 0.124071, 0.149446,
              0.125838, 0.134507, 0.127456, 0.143563, 0.137208, 0.132104,
              0.150874, 0.128177, 0.127039, 0.134079, 0.137161, 0.141534,
              0.138705, 0.127133, 0.143979, 0.119644, 0.138173, 0.137875,
              0.127502, 0.13985 , 0.150958, 0.151042, 0.126087, 0.127661,
              0.146502, 0.133878, 0.131028, 0.140368, 0.155489, 0.120553,
              0.117853, 0.132176, 0.137966, 0.139487, 0.12697 , 0.13998 ,
              0.14023 , 0.130524, 0.138536, 0.173468, 0.134119, 0.135047,
              0.127299, 0.154459, 0.131183, 0.130047, 0.117995, 0.15493 ,
              0.159299, 0.14678 , 0.125124, 0.114106, 0.143049, 0.143355,
              0.163596, 0.133288, 0.135867, 0.148239]])

    • species_zero
      (chain, draw, tube, species)
      float64
      0.1343 1.145 ... 0.1229 1.098

      array([[[[0.134292 , 1.14528  ],
               [0.119757 , 1.27639  ],
               [0.106511 , 1.23213  ],
               ...,
               [0.122804 , 1.19849  ],
               [0.113203 , 1.23223  ],
               [0.101066 , 1.09361  ]],
      
              [[0.13308  , 1.11781  ],
               [0.12164  , 1.23478  ],
               [0.110483 , 1.21598  ],
               ...,
               [0.123072 , 1.12988  ],
               [0.114354 , 1.24013  ],
               [0.107319 , 1.06359  ]],
      
              [[0.107508 , 1.24769  ],
               [0.113404 , 1.22428  ],
               [0.108342 , 1.33478  ],
               ...,
      ...
               ...,
               [0.112888 , 1.24497  ],
               [0.119891 , 1.16724  ],
               [0.113157 , 1.25905  ]],
      
              [[0.0936497, 1.10901  ],
               [0.1149   , 1.12942  ],
               [0.126336 , 1.13351  ],
               ...,
               [0.122085 , 1.15534  ],
               [0.110285 , 1.20729  ],
               [0.121782 , 1.31178  ]],
      
              [[0.137171 , 1.33195  ],
               [0.139115 , 1.07304  ],
               [0.11255  , 1.19714  ],
               ...,
               [0.139124 , 1.33932  ],
               [0.139762 , 1.13867  ],
               [0.122946 , 1.09754  ]]]])

    • sigma_y
      (chain, draw, sigma_y_dim_0)
      float64
      0.103 0.123 ... 0.0853 0.1167

      array([[[0.103018 , 0.123026 ],
              [0.0712653, 0.0867644],
              [0.0840977, 0.0721943],
              [0.0961897, 0.143075 ],
              [0.0999729, 0.0753687],
              [0.0972597, 0.0938447],
              [0.100208 , 0.0943006],
              [0.108581 , 0.101863 ],
              [0.0800381, 0.0944252],
              [0.120121 , 0.102812 ],
              [0.120881 , 0.103784 ],
              [0.131384 , 0.0990181],
              [0.122994 , 0.0978449],
              [0.0963098, 0.117497 ],
              [0.104064 , 0.0852396],
              [0.0901577, 0.0861396],
              [0.0928114, 0.0835343],
              [0.114679 , 0.109278 ],
              [0.110001 , 0.113177 ],
              [0.110218 , 0.117296 ],
      ...
              [0.112118 , 0.0858086],
              [0.0883329, 0.109607 ],
              [0.10067  , 0.0890761],
              [0.119377 , 0.0972628],
              [0.113095 , 0.100517 ],
              [0.0853434, 0.117981 ],
              [0.0979005, 0.109729 ],
              [0.101912 , 0.0987884],
              [0.0996906, 0.0962712],
              [0.0995598, 0.11588  ],
              [0.133378 , 0.0882827],
              [0.139954 , 0.0885941],
              [0.0786867, 0.115797 ],
              [0.129184 , 0.0858769],
              [0.0780372, 0.111932 ],
              [0.127517 , 0.0986251],
              [0.118874 , 0.0904353],
              [0.0835404, 0.110043 ],
              [0.0980234, 0.11287  ],
              [0.0853034, 0.11668  ]]])

    • mu_max
      (chain, draw, strain)
      float64
      0.1261 0.179 ... 0.2791 0.1866

      array([[[0.126069 , 0.179048 , 0.184824 , 0.192322 ],
              [0.180078 , 0.210761 , 0.190806 , 0.188492 ],
              [0.164546 , 0.185148 , 0.19693  , 0.128054 ],
              [0.137343 , 0.199604 , 0.211589 , 0.124495 ],
              [0.125968 , 0.175851 , 0.227886 , 0.13659  ],
              [0.123487 , 0.16419  , 0.23962  , 0.137194 ],
              [0.112036 , 0.173254 , 0.227916 , 0.139841 ],
              [0.246459 , 0.225569 , 0.176977 , 0.195041 ],
              [0.175933 , 0.143873 , 0.119561 , 0.11164  ],
              [0.117974 , 0.161518 , 0.176947 , 0.19736  ],
              [0.15323  , 0.229529 , 0.260394 , 0.281407 ],
              [0.248116 , 0.0942841, 0.150236 , 0.176903 ],
              [0.297825 , 0.617171 , 0.303841 , 0.249703 ],
              [0.182541 , 0.257677 , 0.185641 , 0.162894 ],
              [0.167486 , 0.186741 , 0.160687 , 0.157819 ],
              [0.232915 , 0.200007 , 0.224811 , 0.216628 ],
              [0.183705 , 0.224647 , 0.212629 , 0.23894  ],
              [0.205838 , 0.208377 , 0.300613 , 0.172045 ],
              [0.235238 , 0.126608 , 0.130119 , 0.157003 ],
              [0.110347 , 0.209053 , 0.21061  , 0.164055 ],
      ...
              [0.213249 , 0.186485 , 0.232683 , 0.198172 ],
              [0.111279 , 0.0971822, 0.135143 , 0.0971543],
              [0.122738 , 0.108125 , 0.13515  , 0.104639 ],
              [0.152356 , 0.121661 , 0.169268 , 0.139303 ],
              [0.140906 , 0.10585  , 0.142514 , 0.113313 ],
              [0.140025 , 0.112564 , 0.144164 , 0.128782 ],
              [0.199587 , 0.233568 , 0.176151 , 0.16114  ],
              [0.155744 , 0.0855396, 0.180768 , 0.139943 ],
              [0.15719  , 0.140236 , 0.173893 , 0.247942 ],
              [0.173768 , 0.121402 , 0.174133 , 0.23292  ],
              [0.141946 , 0.185727 , 0.115014 , 0.104845 ],
              [0.145858 , 0.194746 , 0.123883 , 0.106786 ],
              [0.295072 , 0.388067 , 0.239946 , 0.255198 ],
              [0.141644 , 0.183303 , 0.125389 , 0.117299 ],
              [0.137285 , 0.088637 , 0.16025  , 0.1753   ],
              [0.195335 , 0.292827 , 0.160752 , 0.150263 ],
              [0.141828 , 0.196778 , 0.197706 , 0.102358 ],
              [0.123748 , 0.163135 , 0.135504 , 0.103688 ],
              [0.142825 , 0.188884 , 0.151832 , 0.109062 ],
              [0.205775 , 0.233266 , 0.279075 , 0.18658  ]]])

    • ks
      (chain, draw, strain)
      float64
      0.3584 0.1935 ... 0.2787 0.2555

      array([[[0.358394 , 0.19354  , 0.35639  , 0.340078 ],
              [0.167331 , 0.394472 , 0.14812  , 0.157276 ],
              [0.236434 , 0.221749 , 0.256794 , 0.270389 ],
              [0.189216 , 0.237706 , 0.304754 , 0.225537 ],
              [0.15434  , 0.220281 , 0.252545 , 0.199427 ],
              [0.197394 , 0.229662 , 0.279733 , 0.230203 ],
              [0.172353 , 0.22992  , 0.301982 , 0.230139 ],
              [0.26328  , 0.232282 , 0.251349 , 0.252023 ],
              [0.318689 , 0.167896 , 0.252545 , 0.198521 ],
              [0.255973 , 0.214924 , 0.387807 , 0.364625 ],
              [0.256716 , 0.204376 , 0.328749 , 0.339381 ],
              [0.220202 , 0.225177 , 0.191394 , 0.176607 ],
              [0.401454 , 0.296197 , 0.365716 , 0.233325 ],
              [0.311033 , 0.407301 , 0.343247 , 0.186663 ],
              [0.276739 , 0.370266 , 0.297682 , 0.153298 ],
              [0.210184 , 0.154422 , 0.175222 , 0.430208 ],
              [0.268702 , 0.424053 , 0.472738 , 0.117193 ],
              [0.433536 , 0.3474   , 0.187022 , 0.252694 ],
              [0.299349 , 0.249    , 0.2966   , 0.324397 ],
              [0.240343 , 0.525298 , 0.47474  , 0.337256 ],
      ...
              [0.128762 , 0.227553 , 0.26397  , 0.226377 ],
              [0.273468 , 0.406108 , 0.446107 , 0.399844 ],
              [0.119614 , 0.228168 , 0.277254 , 0.23206  ],
              [0.200576 , 0.357892 , 0.380529 , 0.369944 ],
              [0.145705 , 0.299927 , 0.307749 , 0.308327 ],
              [0.170547 , 0.349503 , 0.32225  , 0.353155 ],
              [0.538952 , 0.220622 , 0.290582 , 0.257943 ],
              [0.232064 , 0.505052 , 0.281351 , 0.390628 ],
              [0.274454 , 0.339813 , 0.182059 , 0.201638 ],
              [0.252437 , 0.333337 , 0.153369 , 0.175558 ],
              [0.190691 , 0.19454  , 0.305143 , 0.354961 ],
              [0.19201  , 0.192228 , 0.316143 , 0.365716 ],
              [0.221597 , 0.191003 , 0.45929  , 0.582018 ],
              [0.198414 , 0.166255 , 0.352683 , 0.37479  ],
              [0.406497 , 0.463154 , 0.238207 , 0.2108   ],
              [0.253604 , 0.224043 , 0.426166 , 0.492228 ],
              [0.316609 , 0.541361 , 0.36573  , 0.297143 ],
              [0.332118 , 0.474984 , 0.313404 , 0.280895 ],
              [0.302243 , 0.409423 , 0.288323 , 0.279869 ],
              [0.234427 , 0.341377 , 0.278705 , 0.255511 ]]])

    • gamma
      (chain, draw, strain)
      float64
      0.7393 0.5507 ... 0.5374 0.6419

      array([[[0.739349, 0.550688, 0.618225, 0.683912],
              [0.424732, 0.563855, 0.490083, 0.450982],
              [0.51467 , 0.58417 , 0.575714, 0.589333],
              [0.572909, 0.590288, 0.516478, 0.56663 ],
              [0.646929, 0.694821, 0.577138, 0.65786 ],
              [0.572655, 0.617284, 0.526388, 0.601311],
              [0.59917 , 0.628656, 0.560935, 0.601013],
              [0.526827, 0.688478, 0.663884, 0.546945],
              [0.549171, 0.684162, 0.642259, 0.523473],
              [0.620893, 0.447831, 0.473764, 0.630052],
              [0.607187, 0.407273, 0.450368, 0.660511],
              [0.643745, 0.782215, 0.595321, 0.638368],
              [0.582144, 0.555999, 0.561642, 0.637995],
              [0.546588, 0.527848, 0.558884, 0.784514],
              [0.638573, 0.594135, 0.661748, 0.782198],
              [0.635589, 0.74068 , 0.623461, 0.551169],
              [0.716592, 0.552966, 0.650653, 0.785385],
              [0.527506, 0.509746, 0.891344, 0.537352],
              [0.639624, 0.502379, 0.484911, 0.541865],
              [0.682942, 0.622431, 0.580514, 0.562991],
      ...
              [0.646984, 0.719403, 0.565612, 0.545535],
              [0.671359, 0.758894, 0.597549, 0.545813],
              [0.620512, 0.665632, 0.545014, 0.507084],
              [0.71132 , 0.766935, 0.597586, 0.540899],
              [0.51742 , 0.558166, 0.439566, 0.418916],
              [0.669523, 0.760282, 0.544109, 0.534507],
              [0.587363, 0.519306, 0.623034, 0.613743],
              [0.541361, 0.667242, 0.535707, 0.562304],
              [0.500132, 0.566578, 0.723239, 0.597144],
              [0.509115, 0.608302, 0.772702, 0.629063],
              [0.650054, 0.659149, 0.552983, 0.551946],
              [0.52796 , 0.541337, 0.453867, 0.448303],
              [0.638621, 0.622707, 0.51997 , 0.488685],
              [0.614798, 0.584579, 0.473218, 0.473063],
              [0.648858, 0.6899  , 0.862008, 0.886299],
              [0.704752, 0.654733, 0.530587, 0.508636],
              [0.658806, 0.787578, 0.729392, 0.636307],
              [0.755005, 0.811794, 0.728254, 0.651145],
              [0.629119, 0.674015, 0.582069, 0.550966],
              [0.72196 , 0.736301, 0.537399, 0.641923]]])

    • abundance
      (chain, draw, tube, timepoint, species)
      float64
      0.1344 1.145 0.145 ... 1.063 0.4942

      array([[[[[0.134421 , 1.14518  ],
                [0.14499  , 1.13737  ],
                [0.156371 , 1.12895  ],
                ...,
                [0.472796 , 0.895002 ],
                [0.507448 , 0.869383 ],
                [0.544299 , 0.842137 ]],
      
               [[0.119944 , 1.27629  ],
                [0.135589 , 1.26768  ],
                [0.153258 , 1.25795  ],
                ...,
                [0.92695  , 0.831882 ],
                [1.03865  , 0.77037  ],
                [1.16159  , 0.702669 ]],
      
               [[0.106664 , 1.23203  ],
                [0.119427 , 1.22414  ],
                [0.133694 , 1.21532  ],
                ...,
      ...
                ...,
                [1.39665  , 0.413406 ],
                [1.53516  , 0.311414 ],
                [1.66293  , 0.217339 ]],
      
               [[0.140076 , 1.1385   ],
                [0.167142 , 1.12396  ],
                [0.19934  , 1.10665  ],
                ...,
                [1.99388  , 0.142269 ],
                [2.11829  , 0.075412 ],
                [2.19374  , 0.0348663]],
      
               [[0.123132 , 1.09742  ],
                [0.138735 , 1.0874   ],
                [0.156279 , 1.07614  ],
                ...,
                [0.86867  , 0.618842 ],
                [0.962576 , 0.558562 ],
                [1.06284  , 0.494198 ]]]]])

  • Indexes: (10)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99],
            dtype='int64', name='draw'))

    • ln_mu_max_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_mu_max_z_dim_0'))

    • ln_ks_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_ks_z_dim_0'))

    • ln_gamma_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_gamma_z_dim_0'))

    • tube
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], dtype='int64', name='tube'))

    • species
      PandasIndex

      PandasIndex(Index([1, 2], dtype='int64', name='species'))

    • sigma_y_dim_0
      PandasIndex

      PandasIndex(Index([0, 1], dtype='int64', name='sigma_y_dim_0'))

    • strain
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4], dtype='int64', name='strain'))

    • timepoint
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], dtype='int64', name='timepoint'))

  • Attributes: (4)

    created_at :

    2024-04-30T07:25:50.742269

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• posterior_predictive

  <xarray.Dataset> Size: 130kB
  Dimensions:      (chain: 1, draw: 100, measurement: 160)
  Coordinates:
    * chain        (chain) int64 8B 0
    * draw         (draw) int64 800B 0 1 2 3 4 5 6 7 8 ... 92 93 94 95 96 97 98 99
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      yrep         (chain, draw, measurement) float64 128kB 0.1959 1.416 ... 0.666
  Attributes:
      created_at:                 2024-04-30T07:25:50.748438
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 100
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 6 ... 94 95 96 97 98 99

      array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • yrep
      (chain, draw, measurement)
      float64
      0.1959 1.416 0.1885 ... 0.806 0.666

      array([[[0.195866, 1.41642 , 0.188514, ..., 0.747434, 0.586974,
               1.00298 ],
              [0.160048, 1.0563  , 0.286886, ..., 0.859076, 0.85545 ,
               0.776678],
              [0.135647, 1.31836 , 0.192262, ..., 0.829057, 0.435652,
               0.884375],
              ...,
              [0.166325, 1.21941 , 0.226758, ..., 1.14687 , 0.302638,
               0.968309],
              [0.14274 , 1.18061 , 0.155557, ..., 1.13299 , 0.418849,
               1.20289 ],
              [0.215505, 1.41985 , 0.317938, ..., 0.745954, 0.805973,
               0.666039]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99],
            dtype='int64', name='draw'))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-30T07:25:50.748438

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• log_likelihood

  <xarray.Dataset> Size: 130kB
  Dimensions:      (chain: 1, draw: 100, measurement: 160)
  Coordinates:
    * chain        (chain) int64 8B 0
    * draw         (draw) int64 800B 0 1 2 3 4 5 6 7 8 ... 92 93 94 95 96 97 98 99
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      llik         (chain, draw, measurement) float64 128kB -1.07 0.1169 ... -1.94
  Attributes:
      created_at:                 2024-04-30T07:25:50.749531
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 100
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 6 ... 94 95 96 97 98 99

      array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • llik
      (chain, draw, measurement)
      float64
      -1.07 0.1169 -14.19 ... 1.676 -1.94

      array([[[-1.06968e+00,  1.16949e-01, -1.41851e+01, ...,  9.51414e-01,
               -3.02032e-01,  2.51154e-03],
              [ 9.03113e-01, -8.86681e-01, -6.97452e+00, ...,  8.62151e-01,
                9.51545e-01, -1.21425e+00],
              [-8.83552e+00,  1.04717e+00, -2.40412e+01, ..., -3.97020e+00,
               -1.88287e+01,  1.78695e+00],
              ...,
              [-1.43600e+00,  9.59053e-01, -1.87099e+01, ..., -6.93967e+00,
               -4.66261e+01, -4.11203e-01],
              [-1.64705e+01, -2.89701e-01, -3.74528e+01, ..., -8.23424e+00,
               -2.26718e+01, -1.17466e+00],
              [ 2.59532e+00,  9.46167e-01, -2.12880e-01, ...,  1.41127e+00,
                1.67558e+00, -1.93967e+00]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99],
            dtype='int64', name='draw'))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-30T07:25:50.749531

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• sample_stats

  <xarray.Dataset> Size: 11kB
  Dimensions:          (chain: 1, draw: 200)
  Coordinates:
    * chain            (chain) int64 8B 0
    * draw             (draw) int64 2kB 0 1 2 3 4 5 6 ... 194 195 196 197 198 199
  Data variables:
      lp               (chain, draw) float64 2kB -58.6 -58.6 ... -56.34 -51.74
      acceptance_rate  (chain, draw) float64 2kB 0.9268 0.0 0.0 ... 0.6867 0.9951
      step_size        (chain, draw) float64 2kB 0.03125 12.59 ... 0.05727 0.05727
      tree_depth       (chain, draw) int64 2kB 7 0 0 4 6 7 7 4 ... 4 4 6 6 6 4 5 5
      n_steps          (chain, draw) int64 2kB 127 1 1 15 95 ... 63 63 31 39 47
      diverging        (chain, draw) bool 200B False True True ... False False
      energy           (chain, draw) float64 2kB 78.86 82.76 80.81 ... 80.22 71.82
  Attributes:
      created_at:                 2024-04-30T07:25:50.746660
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 200
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 195 196 197 198 199

      array([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,
              14,  15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,
              28,  29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,
              42,  43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,
              56,  57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,
              70,  71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,
              84,  85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,
              98,  99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
             112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
             126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139,
             140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153,
             154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167,
             168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
             182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195,
             196, 197, 198, 199])

  • Data variables: (7)
    • lp
      (chain, draw)
      float64
      -58.6 -58.6 -58.6 ... -56.34 -51.74

      array([[-58.6048, -58.6048, -58.6048, -58.6048, -57.0495, -60.0722,
              -55.2909, -60.2785, -63.1716, -61.9213, -69.2293, -62.5434,
              -60.4092, -58.1287, -54.828 , -53.7938, -52.5055, -58.7889,
              -56.6883, -58.6782, -57.0165, -58.8267, -58.6879, -58.0898,
              -51.8262, -63.2264, -55.4352, -54.0964, -65.8154, -59.5635,
              -59.607 , -67.2881, -58.1025, -57.4409, -59.2845, -56.3952,
              -61.8801, -67.7182, -65.7617, -60.9002, -61.8904, -63.4106,
              -67.6167, -55.6828, -57.7476, -59.8308, -62.0655, -68.1531,
              -71.9688, -65.5754, -57.5692, -66.398 , -56.8265, -54.7328,
              -60.9109, -60.4608, -65.3202, -52.868 , -57.2536, -55.3238,
              -53.1337, -53.5083, -57.6754, -65.0288, -57.7689, -59.108 ,
              -65.6542, -56.7001, -54.316 , -54.5284, -61.4435, -62.0594,
              -65.4503, -63.9036, -57.9646, -61.1691, -58.67  , -59.9672,
              -62.2866, -67.5737, -51.7767, -53.291 , -48.9425, -51.3255,
              -57.009 , -50.455 , -51.0243, -52.9944, -66.1919, -51.9835,
              -51.6539, -60.3069, -51.2788, -52.8907, -58.3601, -59.8951,
              -55.1466, -54.9873, -54.6765, -59.5826, -57.1969, -59.3743,
              -56.6196, -58.8966, -62.1715, -61.0625, -56.7016, -54.1262,
              -56.868 , -64.3248, -69.0715, -54.176 , -61.686 , -59.8144,
              -59.4087, -58.3806, -60.8654, -65.4199, -64.8562, -65.2462,
              -70.6903, -62.4129, -55.9886, -61.3174, -61.5457, -61.2949,
              -56.0023, -59.7949, -61.993 , -69.2833, -70.6813, -60.5745,
              -69.1824, -57.692 , -70.8145, -57.9204, -58.3472, -58.6702,
              -55.9889, -56.6111, -52.3047, -50.2385, -56.6622, -60.6632,
              -54.4951, -54.0938, -65.008 , -58.8823, -54.3016, -52.894 ,
              -52.8059, -52.4381, -58.1012, -67.1034, -56.6691, -56.0413,
              -60.7917, -61.3587, -58.0845, -56.7403, -59.7203, -61.7424,
              -68.7743, -77.8869, -59.36  , -66.8403, -63.7877, -68.7899,
              -68.8369, -70.6389, -71.6705, -67.2872, -57.1933, -57.6935,
              -61.9538, -58.3722, -57.9886, -60.1303, -61.9579, -60.267 ,
              -64.199 , -69.2978, -73.0657, -68.176 , -73.3675, -65.5747,
              -58.9866, -61.9784, -57.6639, -59.3372, -62.3391, -65.8443,
              -75.5784, -60.819 , -60.0743, -59.7531, -58.6591, -52.5399,
              -56.3384, -51.7446]])

    • acceptance_rate
      (chain, draw)
      float64
      0.9268 0.0 0.0 ... 0.6867 0.9951

      array([[9.26757e-01, 0.00000e+00, 0.00000e+00, 4.15275e-07, 9.76496e-01,
              9.67671e-01, 8.80693e-01, 9.05453e-01, 8.71827e-01, 9.17820e-01,
              7.73219e-01, 9.17951e-01, 6.04554e-01, 9.96156e-01, 6.30245e-01,
              4.94417e-01, 9.67753e-01, 7.90591e-01, 9.91040e-01, 8.30714e-01,
              6.48149e-01, 6.87872e-01, 9.86418e-01, 9.84442e-01, 7.06031e-02,
              9.85406e-01, 9.98580e-01, 5.34236e-01, 8.50447e-01, 9.98506e-01,
              7.86500e-01, 6.82376e-01, 1.00000e+00, 8.27157e-01, 9.54891e-01,
              8.17556e-02, 9.53133e-01, 9.91099e-01, 8.61260e-01, 7.69243e-01,
              9.03339e-01, 5.82686e-01, 8.00382e-01, 9.39302e-01, 2.91911e-01,
              9.81353e-01, 1.00000e+00, 9.15554e-01, 3.33310e-01, 9.70604e-01,
              9.96376e-01, 5.04699e-01, 9.94489e-01, 5.12153e-01, 8.72913e-01,
              9.32881e-01, 3.91760e-01, 1.00000e+00, 9.32069e-01, 7.72947e-01,
              9.76212e-01, 3.17392e-01, 9.60516e-01, 8.48580e-01, 1.00000e+00,
              4.68695e-01, 8.75956e-01, 8.84637e-01, 7.47462e-01, 7.74819e-01,
              7.83783e-01, 7.66883e-01, 8.17644e-01, 8.25479e-01, 9.88187e-01,
              4.04629e-01, 9.28758e-01, 5.93088e-01, 9.38169e-01, 8.20680e-01,
              1.00000e+00, 2.73707e-01, 9.99757e-01, 5.35283e-01, 9.15210e-01,
              9.99319e-01, 6.94933e-01, 9.00822e-01, 5.05535e-01, 9.98875e-01,
              7.74495e-01, 8.19984e-01, 9.97671e-01, 4.89215e-01, 9.01405e-01,
              8.58887e-01, 7.61627e-01, 5.16337e-01, 8.68469e-01, 8.48006e-01,
              6.53246e-01, 8.75766e-01, 9.31120e-01, 7.70085e-01, 8.67168e-01,
              9.75150e-01, 8.24386e-01, 9.92946e-01, 7.27563e-01, 8.07874e-01,
              8.81715e-01, 9.97891e-01, 7.03539e-01, 6.93359e-01, 8.39117e-01,
              9.97535e-01, 8.40310e-01, 9.70479e-01, 8.76601e-01, 9.63590e-01,
              7.80645e-01, 9.96861e-01, 9.27210e-01, 9.10064e-01, 9.50023e-01,
              9.77658e-01, 7.47987e-01, 5.67010e-01, 9.03358e-01, 9.18203e-01,
              8.14216e-01, 9.64671e-01, 7.83933e-01, 9.79917e-01, 5.48645e-01,
              9.99840e-01, 8.61332e-01, 8.07722e-01, 9.42404e-01, 8.37435e-01,
              9.96729e-01, 8.05703e-01, 8.04326e-01, 9.76829e-01, 8.33988e-01,
              8.83372e-01, 7.41326e-01, 9.91197e-01, 9.81908e-01, 9.56325e-01,
              8.83673e-01, 8.81153e-01, 8.32485e-01, 7.30035e-01, 1.00000e+00,
              9.97318e-01, 9.93111e-01, 9.97750e-01, 8.67459e-01, 7.44329e-01,
              6.86905e-01, 7.61603e-01, 8.49242e-01, 6.90757e-01, 1.00000e+00,
              6.69544e-01, 9.84007e-01, 7.02282e-01, 9.78342e-01, 7.31150e-01,
              8.94235e-01, 9.85702e-01, 9.99967e-01, 9.38881e-01, 8.61789e-01,
              9.71803e-01, 7.87307e-01, 7.11989e-01, 9.70108e-01, 9.29145e-01,
              7.27099e-01, 8.08279e-01, 8.74393e-01, 9.99637e-01, 7.81144e-01,
              1.00000e+00, 9.43050e-01, 8.46536e-01, 9.48579e-01, 9.17649e-01,
              9.12324e-01, 8.60802e-01, 8.98609e-01, 1.00000e+00, 7.82472e-01,
              9.75986e-01, 9.43740e-01, 9.97829e-01, 6.86744e-01, 9.95121e-01]])

    • step_size
      (chain, draw)
      float64
      0.03125 12.59 ... 0.05727 0.05727

      array([[ 0.03125  , 12.5919   ,  2.04569  ,  0.197275 ,  0.0151101,
               0.0192796,  0.0272152,  0.0317209,  0.0417976,  0.0510688,
               0.0735843,  0.0679933,  0.100071 ,  0.0553381,  0.104964 ,
               0.0638403,  0.0258198,  0.0450416,  0.0456492,  0.0846505,
               0.0964796,  0.0640191,  0.0480947,  0.0865242,  0.153314 ,
               0.0197702,  0.0352499,  0.0645981,  0.032157 ,  0.0387887,
               0.0699669,  0.0702711,  0.0533096,  0.0949123,  0.105883 ,
               0.164967 ,  0.0261627,  0.0407291,  0.0694148,  0.0842709,
               0.0808117,  0.108571 ,  0.0652963,  0.0677885,  0.0989324,
               0.0295992,  0.0478377,  0.0803101,  0.109395 ,  0.0372666,
               0.0578346,  0.0947681,  0.049153 ,  0.0797376,  0.0424601,
               0.0519046,  0.072498 ,  0.0297821,  0.0482974,  0.0669265,
               0.0649488,  0.0985937,  0.0352479,  0.0516292,  0.0590995,
               0.0935246,  0.0472792,  0.0572374,  0.0704191,  0.0647714,
               0.0631338,  0.0627051,  0.0601354,  0.0640441,  0.0692567,
               0.10424  ,  0.0478571,  0.0637348,  0.0431427,  0.0583971,
               0.0623914,  0.0949678,  0.034459 ,  0.0523532,  0.0320236,
               0.0411334,  0.0620068,  0.0518535,  0.0644112,  0.0375903,
               0.0562021,  0.0547981,  0.0582087,  0.0861876,  0.0492809,
               0.0608541,  0.0693346,  0.0659688,  0.0400478,  0.0464272,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658,
               0.0572658,  0.0572658,  0.0572658,  0.0572658,  0.0572658]])

    • tree_depth
      (chain, draw)
      int64
      7 0 0 4 6 7 7 4 ... 4 4 6 6 6 4 5 5

      array([[7, 0, 0, 4, 6, 7, 7, 4, 7, 5, 5, 6, 6, 3, 4, 3, 4, 3, 6, 6, 5, 3,
              6, 5, 5, 7, 7, 4, 5, 7, 6, 6, 3, 5, 5, 5, 5, 6, 6, 6, 6, 5, 6, 6,
              3, 4, 5, 6, 5, 6, 5, 4, 3, 4, 7, 5, 5, 4, 6, 6, 5, 4, 7, 3, 3, 5,
              4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 5, 7, 6, 4, 3, 6, 3, 4, 7, 4, 6,
              5, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 6, 3, 6, 6, 4, 4, 3, 3, 5, 4, 6,
              3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 3, 6, 6, 4, 6, 6, 6, 3, 6,
              6, 6, 3, 6, 6, 5, 4, 4, 4, 3, 4, 6, 3, 5, 5, 4, 3, 6, 6, 6, 5, 4,
              3, 4, 6, 6, 6, 6, 6, 6, 6, 3, 3, 6, 6, 3, 3, 6, 3, 6, 6, 5, 6, 5,
              3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 6, 6, 6, 4, 6, 4, 4, 4, 6, 6, 6, 4,
              5, 5]])

    • n_steps
      (chain, draw)
      int64
      127 1 1 15 95 ... 63 63 31 39 47

      array([[127,   1,   1,  15,  95, 127, 127,  15, 127,  39,  31,  63, 127,
                7,  31,  15,  15,  15,  63,  63,  31,  15,  63,  31,  31, 255,
              127,  23,  47, 127,  63,  63,   7,  31,  31,  31,  63,  63,  63,
               63,  63,  31,  63,  63,  15,  15,  47,  63,  31,  63,  55,  31,
               15,  23, 127,  47,  63,  15,  63,  63,  47,  31, 127,  15,   7,
               31,  15,  63,  63,  63,  63,  63,  63,  63,  63,  31,  63,  39,
              127,  63,  23,   7,  95,   7,  31, 127,  31,  63,  39,  63,  47,
               63,  63,  47,  63,  63,  63,  63, 127, 127,  15,  63,  63,  31,
               23,  15,  15,  39,  23,  63,   7,  63,  63,  63,  63,  63,  63,
               63,  63,  63,  63,  63,  55,   7,  63,  63,  23,  63,  63,  63,
                7,  63,  63,  63,   7,  63,  63,  55,  31,  31,  31,  15,  31,
               63,  15,  39,  39,  31,  15,  63,  63,  63,  47,  23,   7,  31,
               63,  63,  63,  63,  63,  63,  63,   7,   7,  63,  63,  15,   7,
               63,   7,  63,  63,  47,  63,  55,  15,  23,  23,  15,   7,   7,
                7,  23,  23,   7,  63,  63,  63,  15,  63,  23,  23,  15,  63,
               63,  63,  31,  39,  47]])

    • diverging
      (chain, draw)
      bool
      False True True ... False False

      array([[False,  True,  True, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False]])

    • energy
      (chain, draw)
      float64
      78.86 82.76 80.81 ... 80.22 71.82

      array([[ 78.8602,  82.7591,  80.8114,  80.7794,  82.7418,  91.0099,
               92.9877,  80.0374,  86.5825,  91.9513,  93.7189,  92.7244,
               86.8454,  79.5793,  80.4784,  83.376 ,  73.1065,  74.3291,
               80.9646,  79.6666,  80.1301,  82.3769,  77.5666,  75.6883,
               87.3459,  82.7512,  81.888 ,  83.0343,  85.6575,  92.1114,
               84.802 ,  87.1944,  82.4074,  77.7009,  78.5077,  87.468 ,
               89.3935,  98.349 ,  95.5323,  92.6367,  89.1492,  88.5632,
               91.0312,  88.5949,  94.8072,  81.2396,  82.8426,  89.4432,
              110.435 , 103.165 ,  88.1175,  89.0558,  86.6067,  84.5048,
               78.1522,  89.1505, 103.437 ,  78.7436,  76.7479,  80.5473,
               72.673 ,  81.7957,  77.9485,  84.769 ,  83.4912,  87.7797,
               83.8309,  94.1106,  81.6622,  78.2644,  81.5752,  89.0636,
               93.3087,  97.4376,  86.3968,  87.4867,  85.599 ,  94.9595,
               91.2701,  90.7952,  83.1709,  79.3676,  67.6398,  86.6999,
               75.5488,  72.6459,  76.9973,  76.5047,  86.1134,  84.0877,
               83.8123,  77.1836,  76.8308,  77.045 ,  76.4716,  82.9793,
               82.967 ,  84.9756,  79.2393,  82.7648,  88.1495,  83.5465,
               79.858 ,  86.5232,  84.4287,  82.4817,  85.1366,  74.8682,
               84.5242,  84.8963,  92.5777,  86.8654,  88.0376,  93.2229,
               85.4782,  83.2684,  85.7699,  88.3162,  91.5251,  93.5257,
              102.933 ,  95.3181,  86.069 ,  77.9593,  87.4147,  87.0359,
               88.2101,  90.0936,  82.5068,  85.4613,  95.8255,  94.065 ,
               91.0461,  94.1312,  90.5257,  87.385 ,  85.7329,  85.2045,
               85.4298,  83.0275,  74.004 ,  74.9827,  76.6576,  80.1785,
               88.6778,  75.6836,  84.5215,  82.8775,  76.9601,  76.5793,
               72.4169,  70.8732,  72.974 ,  91.6481,  87.4817,  77.3688,
               82.2084,  80.5267,  86.4446,  86.034 ,  93.1437,  88.4245,
               95.3553, 106.628 ,  96.2024,  95.129 ,  87.1269,  93.9725,
               91.2743, 109.742 ,  97.3843,  94.5153,  82.8333,  89.2644,
               86.2935,  82.8175,  87.0762,  90.5778,  88.8285,  81.3797,
               86.2099,  92.1065,  99.421 , 102.038 , 107.95  ,  93.8631,
               89.7448,  88.4546,  86.5707,  81.8977,  86.4589,  88.9054,
               97.7194,  94.3079,  90.2654,  82.3723,  80.3507,  77.1395,
               80.2215,  71.82  ]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             190, 191, 192, 193, 194, 195, 196, 197, 198, 199],
            dtype='int64', name='draw', length=200))

  • Attributes: (4)

    created_at :

    2024-04-30T07:25:50.746660

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  

We can find the prior intervals for the true abundance and plot them in the graph.

prior_abundances = idata_prior.posterior["abundance"]

n_sample = 20
chains = rng.choice(prior_abundances.coords["chain"].values, n_sample)
draws = rng.choice(prior_abundances.coords["draw"].values, n_sample)
f, axes = plot_sim(true_abundance, fake_measurements, species_to_ax)

for ax, species_i in zip(axes, species):
    for tube_j in tubes:
        for chain, draw in zip(chains, draws):
            timeseries = prior_abundances.sel(chain=chain, draw=draw, tube=tube_j, species=species_i)
            ax.plot(
                timepoints.values, 
                timeseries.values,
                alpha=0.5, color="skyblue", zorder=-1
            )
f.savefig("img/monod_priors.png")

Sample in posterior mode

stan_input_posterior = stan_input_common | {
    "y": fake_measurements["simulated_measurement"],
    "likelihood": 1
}
mcmc_posterior = model.sample(
    data=stan_input_posterior,
    iter_warmup=300,
    iter_sampling=300,
    chains=4,
    refresh=1,
    inits=true_param_values,
    seed=SEED,
)
idata_posterior = az.from_cmdstanpy(
    mcmc_posterior,
    dims=dims,
    coords=coords,
    posterior_predictive={"y": "yrep"},
    log_likelihood="llik"
)
idata_posterior

09:25:51 - cmdstanpy - INFO - CmdStan start processing
09:32:24 - cmdstanpy - INFO - CmdStan done processing.
09:32:24 - cmdstanpy - WARNING - Non-fatal error during sampling:
Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
Exception: ode_bdf_tol: initial state[1] is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
Exception: ode_bdf_tol: initial state[1] is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: lognormal_lpdf: Location parameter is -nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
Consider re-running with show_console=True if the above output is unclear!

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 7MB
  Dimensions:            (chain: 4, draw: 300, ln_mu_max_z_dim_0: 4,
                          ln_ks_z_dim_0: 4, ln_gamma_z_dim_0: 4, tube: 16,
                          species: 2, sigma_y_dim_0: 2, strain: 4, timepoint: 20)
  Coordinates:
    * chain              (chain) int64 32B 0 1 2 3
    * draw               (draw) int64 2kB 0 1 2 3 4 5 ... 294 295 296 297 298 299
    * ln_mu_max_z_dim_0  (ln_mu_max_z_dim_0) int64 32B 0 1 2 3
    * ln_ks_z_dim_0      (ln_ks_z_dim_0) int64 32B 0 1 2 3
    * ln_gamma_z_dim_0   (ln_gamma_z_dim_0) int64 32B 0 1 2 3
    * tube               (tube) int64 128B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    * species            (species) int64 16B 1 2
    * sigma_y_dim_0      (sigma_y_dim_0) int64 16B 0 1
    * strain             (strain) int64 32B 1 2 3 4
    * timepoint          (timepoint) int64 160B 1 2 3 4 5 6 ... 15 16 17 18 19 20
  Data variables: (12/15)
      ln_mu_max_z        (chain, draw, ln_mu_max_z_dim_0) float64 38kB 1.44 ......
      ln_ks_z            (chain, draw, ln_ks_z_dim_0) float64 38kB 0.2322 ... 0...
      ln_gamma_z         (chain, draw, ln_gamma_z_dim_0) float64 38kB -0.4819 ....
      a_mu_max           (chain, draw) float64 10kB -1.552 -1.759 ... -1.551 -1.77
      a_ks               (chain, draw) float64 10kB -1.211 -1.338 ... -1.422
      a_gamma            (chain, draw) float64 10kB -0.5341 -0.5231 ... -0.4704
      ...                 ...
      species_zero       (chain, draw, tube, species) float64 307kB 0.127 ... 1...
      sigma_y            (chain, draw, sigma_y_dim_0) float64 19kB 0.08225 ... ...
      mu_max             (chain, draw, strain) float64 38kB 0.3066 ... 0.1696
      ks                 (chain, draw, strain) float64 38kB 0.319 ... 0.2912
      gamma              (chain, draw, strain) float64 38kB 0.5479 ... 0.6565
      abundance          (chain, draw, tube, timepoint, species) float64 6MB 0....
  Attributes:
      created_at:                 2024-04-30T07:32:24.228168
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
    • ln_mu_max_z_dim_0: 4
    • ln_ks_z_dim_0: 4
    • ln_gamma_z_dim_0: 4
    • tube: 16
    • species: 2
    • sigma_y_dim_0: 2
    • strain: 4
    • timepoint: 20
  • Coordinates: (10)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

    • ln_mu_max_z_dim_0
      (ln_mu_max_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_ks_z_dim_0
      (ln_ks_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_gamma_z_dim_0
      (ln_gamma_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • tube
      (tube)
      int64
      1 2 3 4 5 6 7 ... 11 12 13 14 15 16

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16])

    • species
      (species)
      int64
      1 2

      array([1, 2])

    • sigma_y_dim_0
      (sigma_y_dim_0)
      int64
      0 1

      array([0, 1])

    • strain
      (strain)
      int64
      1 2 3 4

      array([1, 2, 3, 4])

    • timepoint
      (timepoint)
      int64
      1 2 3 4 5 6 7 ... 15 16 17 18 19 20

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
             19, 20])

  • Data variables: (15)
    • ln_mu_max_z
      (chain, draw, ln_mu_max_z_dim_0)
      float64
      1.44 -2.075 ... 0.1222 -0.0197

      array([[[ 1.43964  , -2.07539  , -0.480108 , -0.476268 ],
              [ 2.14933  , -0.639818 , -0.136004 ,  0.43     ],
              [ 1.83206  , -1.22235  , -0.115291 ,  0.868755 ],
              ...,
              [ 2.5687   , -1.44086  ,  0.621402 , -0.328346 ],
              [ 2.79695  , -1.60759  ,  0.830904 ,  0.0429799],
              [ 2.66339  , -1.69941  ,  0.80337  , -0.199915 ]],
      
             [[ 1.77155  , -1.70135  , -0.128963 , -0.307049 ],
              [ 1.33849  , -1.35961  , -0.937327 , -0.882081 ],
              [ 1.46541  , -1.0822   , -0.959903 , -1.0799   ],
              ...,
              [ 2.5895   , -0.763728 ,  0.588707 ,  0.720792 ],
              [ 1.78995  , -1.48543  , -0.589971 , -0.429114 ],
              [ 2.68589  , -1.03037  , -0.144811 , -0.0397646]],
      
             [[ 2.80766  , -0.499835 ,  0.856673 ,  1.15522  ],
              [ 2.283    , -0.626296 ,  0.700771 ,  0.782596 ],
              [ 2.36057  , -0.282299 ,  0.656092 ,  0.599962 ],
              ...,
              [ 1.87784  , -1.08237  ,  0.11332  ,  0.147031 ],
              [ 1.8798   , -1.9366   , -0.252805 , -0.561231 ],
              [ 1.86706  , -2.01245  , -0.342733 , -0.402687 ]],
      
             [[ 1.72097  , -0.798292 , -0.0619448,  0.385759 ],
              [ 1.55378  , -1.42811  , -0.0344052, -0.243571 ],
              [ 2.49534  , -0.639839 ,  0.0364284,  0.798284 ],
              ...,
              [ 2.48553  , -0.874164 ,  0.488614 ,  0.345169 ],
              [ 0.935276 , -1.80867  , -0.886871 , -0.649151 ],
              [ 2.23011  , -1.25447  ,  0.122228 , -0.0196958]]])

    • ln_ks_z
      (chain, draw, ln_ks_z_dim_0)
      float64
      0.2322 -0.9934 ... 0.802 0.6242

      array([[[ 0.232193  , -0.993438  ,  1.21535   ,  0.0555106 ],
              [-0.112696  ,  1.30585   , -1.23193   ,  0.760839  ],
              [-0.35623   , -0.21666   , -0.528023  ,  2.01192   ],
              ...,
              [ 0.637344  , -0.447281  ,  2.01426   , -0.905051  ],
              [ 0.479432  , -1.37802   ,  2.12197   , -0.745629  ],
              [ 0.104107  , -0.936886  ,  2.50349   , -0.562207  ]],
      
             [[-0.507451  ,  1.09847   ,  1.39201   ,  0.172546  ],
              [-0.622539  ,  1.77895   , -1.30323   , -1.45846   ],
              [-0.0206634 ,  2.71434   , -0.387853  , -1.38587   ],
              ...,
              [ 0.510413  , -1.41813   ,  0.464166  ,  0.777791  ],
              [-0.778757  ,  0.500842  , -0.439454  ,  0.136273  ],
              [ 1.21993   ,  0.505053  , -0.262018  , -0.910803  ]],
      
             [[-0.20656   , -0.711122  ,  0.471984  ,  0.558205  ],
              [-1.11553   , -0.462498  ,  0.244496  ,  0.204236  ],
              [-0.917741  ,  0.909479  ,  0.223158  , -0.38252   ],
              ...,
              [ 0.266817  ,  1.22857   ,  0.89577   ,  1.37627   ],
              [-0.0507751 , -0.806308  ,  0.182342  , -2.42045   ],
              [ 0.197544  , -2.66593   , -0.760431  , -1.15723   ]],
      
             [[-0.986154  ,  1.17482   , -1.13507   ,  0.784005  ],
              [-1.33177   , -0.967554  ,  0.342234  , -0.0393334 ],
              [-0.194734  ,  1.57597   , -0.427179  ,  0.867152  ],
              ...,
              [-0.0764064 ,  1.03301   ,  1.47118   ,  0.798913  ],
              [-1.60072   , -0.22092   , -1.0382    , -0.64229   ],
              [-0.0859219 ,  0.00345018,  0.802034  ,  0.624204  ]]])

    • ln_gamma_z
      (chain, draw, ln_gamma_z_dim_0)
      float64
      -0.4819 -1.085 ... -1.387 0.3627

      array([[[-0.481869  , -1.08469   ,  1.05593   ,  0.700992  ],
              [-0.376054  , -0.257684  , -0.833916  , -0.437308  ],
              [ 0.430811  , -0.443226  , -1.13387   , -0.665901  ],
              ...,
              [-0.431813  ,  2.97331   ,  0.56932   ,  0.158556  ],
              [-0.899488  ,  2.6287    ,  1.05759   ,  0.186152  ],
              [-0.488969  ,  1.96513   ,  0.581532  ,  0.548718  ]],
      
             [[ 0.556725  ,  0.657734  ,  0.791071  , -0.0964423 ],
              [ 0.306999  ,  0.830107  ,  2.0521    ,  0.316339  ],
              [-0.239339  , -0.152881  ,  1.44206   ,  1.03567   ],
              ...,
              [-0.858468  ,  0.502376  ,  0.165161  , -0.464223  ],
              [ 0.544835  , -0.4659    , -0.153035  ,  2.89561   ],
              [-0.481186  , -2.0996    ,  0.630994  ,  1.10187   ]],
      
             [[-0.660161  ,  0.133659  , -1.06296   , -0.212492  ],
              [-0.77063   ,  0.128364  , -0.75639   , -0.138919  ],
              [-0.18801   ,  0.457842  , -1.1757    , -0.458183  ],
              ...,
              [-0.0124985 ,  0.843747  , -0.119322  ,  1.3118    ],
              [ 0.337861  ,  0.422073  ,  1.03061   ,  1.56057   ],
              [-0.360246  ,  0.502159  ,  1.04323   ,  1.39688   ]],
      
             [[-0.403849  , -0.00556633,  1.00196   , -0.203046  ],
              [ 0.0559757 ,  1.81596   ,  0.873108  ,  2.51037   ],
              [-0.279078  , -1.6516    , -2.87617   , -0.720547  ],
              ...,
              [-0.98695   , -0.89747   , -1.88445   , -1.45724   ],
              [-0.680618  ,  1.83281   ,  0.246619  ,  1.73185   ],
              [-0.94155   , -0.364602  , -1.38733   ,  0.362687  ]]])

    • a_mu_max
      (chain, draw)
      float64
      -1.552 -1.759 ... -1.551 -1.77

      array([[-1.55191, -1.7591 , -1.78322, ..., -1.72957, -1.78749, -1.73027],
             [-1.61848, -1.59448, -1.57081, ..., -1.9078 , -1.63069, -1.74218],
             [-1.97032, -1.90282, -1.92015, ..., -1.69575, -1.68055, -1.70269],
             [-1.76489, -1.73379, -1.78184, ..., -1.83709, -1.55069, -1.76969]])

    • a_ks
      (chain, draw)
      float64
      -1.211 -1.338 ... -1.169 -1.422

      array([[-1.21088, -1.33753, -1.35064, ..., -1.22029, -1.2802 , -1.16265],
             [-1.19039, -1.36207, -1.32648, ..., -1.31252, -1.21392, -1.17491],
             [-1.34337, -1.25926, -1.27406, ..., -1.36062, -1.30982, -1.4372 ],
             [-1.13844, -1.36224, -1.13885, ..., -1.50129, -1.16884, -1.42218]])

    • a_gamma
      (chain, draw)
      float64
      -0.5341 -0.5231 ... -0.5582 -0.4704

      array([[-0.534115, -0.523109, -0.576391, ..., -0.56432 , -0.524737,
              -0.541015],
             [-0.704445, -0.694588, -0.593268, ..., -0.460145, -0.604671,
              -0.439915],
             [-0.48952 , -0.487978, -0.54914 , ..., -0.631705, -0.709394,
              -0.626348],
             [-0.497183, -0.679197, -0.448852, ..., -0.450101, -0.558203,
              -0.47039 ]])

    • t_mu_max
      (chain, draw)
      float64
      0.2569 0.2375 ... 0.284 0.2326

      array([[0.256876, 0.237519, 0.272467, ..., 0.231494, 0.229587, 0.217673],
             [0.228813, 0.25043 , 0.226689, ..., 0.277648, 0.215129, 0.246671],
             [0.255465, 0.270097, 0.270125, ..., 0.270354, 0.251784, 0.266796],
             [0.296958, 0.271821, 0.234714, ..., 0.220754, 0.283978, 0.232559]])

    • t_ks
      (chain, draw)
      float64
      0.2939 0.3037 ... 0.3008 0.302

      array([[0.293897, 0.303692, 0.329511, ..., 0.264736, 0.272606, 0.279762],
             [0.289435, 0.328538, 0.335106, ..., 0.288878, 0.302213, 0.285777],
             [0.32671 , 0.305674, 0.309111, ..., 0.33276 , 0.23913 , 0.273322],
             [0.311004, 0.303607, 0.267113, ..., 0.308393, 0.300841, 0.302048]])

    • t_gamma
      (chain, draw)
      float64
      0.14 0.1359 0.133 ... 0.1273 0.1367

      array([[0.140044, 0.135942, 0.132985, ..., 0.144458, 0.146196, 0.139162],
             [0.143113, 0.110623, 0.10828 , ..., 0.151485, 0.132237, 0.115262],
             [0.138039, 0.144082, 0.14821 , ..., 0.141077, 0.167302, 0.144753],
             [0.131864, 0.136226, 0.116536, ..., 0.132434, 0.127331, 0.136693]])

    • species_zero
      (chain, draw, tube, species)
      float64
      0.127 1.22 0.1262 ... 0.1262 1.191

      array([[[[0.127043, 1.22033 ],
               [0.126188, 1.22567 ],
               [0.136362, 1.25293 ],
               ...,
               [0.114655, 1.17641 ],
               [0.129273, 1.12277 ],
               [0.117   , 1.20786 ]],
      
              [[0.132279, 1.27465 ],
               [0.117335, 1.21091 ],
               [0.127426, 1.14463 ],
               ...,
               [0.114699, 1.14744 ],
               [0.133546, 1.10304 ],
               [0.119708, 1.12792 ]],
      
              [[0.130351, 1.2414  ],
               [0.125514, 1.21064 ],
               [0.137717, 1.11571 ],
               ...,
      ...
               ...,
               [0.107973, 1.18854 ],
               [0.136203, 1.0985  ],
               [0.127934, 1.07662 ]],
      
              [[0.122512, 1.19423 ],
               [0.120131, 1.25898 ],
               [0.13529 , 1.21686 ],
               ...,
               [0.112599, 1.21695 ],
               [0.128991, 1.11756 ],
               [0.120421, 1.21239 ]],
      
              [[0.135157, 1.19852 ],
               [0.123128, 1.31142 ],
               [0.125693, 1.10325 ],
               ...,
               [0.108356, 1.17946 ],
               [0.139036, 1.05028 ],
               [0.126155, 1.1906  ]]]])

    • sigma_y
      (chain, draw, sigma_y_dim_0)
      float64
      0.08225 0.102 ... 0.09004 0.1129

      array([[[0.0822489, 0.101957 ],
              [0.0908424, 0.105442 ],
              [0.0790525, 0.114561 ],
              ...,
              [0.0797601, 0.100791 ],
              [0.0821422, 0.105244 ],
              [0.0846408, 0.115955 ]],
      
             [[0.0725266, 0.118428 ],
              [0.0856653, 0.107317 ],
              [0.0794435, 0.103784 ],
              ...,
              [0.0797231, 0.10365  ],
              [0.0790131, 0.115766 ],
              [0.0728859, 0.103274 ]],
      
             [[0.0704429, 0.0955872],
              [0.0762201, 0.101404 ],
              [0.0777864, 0.0975832],
              ...,
              [0.0697897, 0.0989847],
              [0.0982069, 0.113484 ],
              [0.0911951, 0.109216 ]],
      
             [[0.0920115, 0.123187 ],
              [0.0781625, 0.118374 ],
              [0.0808318, 0.108663 ],
              ...,
              [0.08323  , 0.110853 ],
              [0.0744598, 0.0994558],
              [0.0900424, 0.112892 ]]])

    • mu_max
      (chain, draw, strain)
      float64
      0.3066 0.1243 ... 0.1753 0.1696

      array([[[0.306635, 0.124304, 0.187264, 0.187449],
              [0.286909, 0.147922, 0.166726, 0.190717],
              [0.276915, 0.12048 , 0.162898, 0.212989],
              ...,
              [0.321445, 0.127057, 0.204801, 0.164379],
              [0.318113, 0.115721, 0.202558, 0.169039],
              [0.31647 , 0.122433, 0.211105, 0.169689]],
      
             [[0.297267, 0.134288, 0.192437, 0.184753],
              [0.283858, 0.144429, 0.16054 , 0.162777],
              [0.289786, 0.162654, 0.167226, 0.162739],
              ...,
              [0.304578, 0.12005 , 0.17476 , 0.181288],
              [0.287762, 0.142238, 0.172456, 0.178528],
              [0.339716, 0.135832, 0.168993, 0.17343 ]],
      
             [[0.285628, 0.1227  , 0.173518, 0.187269],
              [0.276321, 0.125936, 0.180225, 0.184253],
              [0.277344, 0.135823, 0.175009, 0.172375],
              ...,
              [0.304809, 0.136918, 0.189169, 0.190901],
              [0.299019, 0.114388, 0.174784, 0.161724],
              [0.299821, 0.1065  , 0.166271, 0.163633]],
      
             [[0.285408, 0.135071, 0.168085, 0.191985],
              [0.26943 , 0.119793, 0.174969, 0.165298],
              [0.302355, 0.144855, 0.169773, 0.203015],
              ...,
              [0.275712, 0.131327, 0.177422, 0.171892],
              [0.276625, 0.126906, 0.164879, 0.176394],
              [0.286202, 0.127272, 0.175298, 0.169607]]])

    • ks
      (chain, draw, strain)
      float64
      0.319 0.2225 ... 0.3073 0.2912

      array([[[0.318976, 0.222496, 0.42584 , 0.302836],
              [0.253662, 0.390257, 0.180567, 0.330726],
              [0.230381, 0.241223, 0.217702, 0.502735],
              ...,
              [0.34939 , 0.262185, 0.50306 , 0.232261],
              [0.316792, 0.190928, 0.495721, 0.22685 ],
              [0.321898, 0.240568, 0.629856, 0.267154]],
      
             [[0.262563, 0.417925, 0.454985, 0.319676],
              [0.208755, 0.459501, 0.166922, 0.158623],
              [0.263579, 0.659097, 0.233062, 0.166811],
              ...,
              [0.3119  , 0.178675, 0.307761, 0.336946],
              [0.234741, 0.345569, 0.260089, 0.309518],
              [0.437672, 0.3568  , 0.286565, 0.238068]],
      
             [[0.243934, 0.206862, 0.304474, 0.313173],
              [0.201846, 0.24644 , 0.305891, 0.30215 ],
              [0.21061 , 0.370489, 0.299667, 0.248502],
              ...,
              [0.280318, 0.386046, 0.345576, 0.405493],
              [0.266611, 0.222543, 0.281896, 0.15128 ],
              [0.250772, 0.114651, 0.193003, 0.173167]],
      
             [[0.235713, 0.461594, 0.225046, 0.408766],
              [0.170918, 0.190902, 0.284127, 0.253047],
              [0.303957, 0.48778 , 0.285659, 0.403643],
              ...,
              [0.217653, 0.306445, 0.350783, 0.285101],
              [0.191972, 0.290746, 0.22737 , 0.25613 ],
              [0.235009, 0.24144 , 0.307302, 0.291231]]])

    • gamma
      (chain, draw, strain)
      float64
      0.5479 0.5036 ... 0.5168 0.6565

      array([[[0.547935, 0.503576, 0.679609, 0.646653],
              [0.563138, 0.572273, 0.529156, 0.558468],
              [0.595056, 0.529759, 0.48327 , 0.514301],
              ...,
              [0.534353, 0.873891, 0.617499, 0.581924],
              [0.518799, 0.868983, 0.69065 , 0.608035],
              [0.543861, 0.765257, 0.631228, 0.628352]],
      
             [[0.535384, 0.54318 , 0.553645, 0.487606],
              [0.516527, 0.547299, 0.626518, 0.517061],
              [0.538384, 0.543447, 0.645892, 0.618087],
              ...,
              [0.554222, 0.681102, 0.647183, 0.588329],
              [0.587063, 0.513616, 0.535311, 0.801104],
              [0.609341, 0.505645, 0.692681, 0.731315]],
      
             [[0.559536, 0.624334, 0.529274, 0.595203],
              [0.549354, 0.625325, 0.550482, 0.601701],
              [0.561578, 0.61799 , 0.485105, 0.539535],
              ...,
              [0.530748, 0.598894, 0.522809, 0.639774],
              [0.52055 , 0.527936, 0.584516, 0.638709],
              [0.50738 , 0.574842, 0.621675, 0.654328]],
      
             [[0.576698, 0.607796, 0.694155, 0.592173],
              [0.510905, 0.649329, 0.571062, 0.713752],
              [0.617933, 0.526595, 0.456563, 0.586946],
              ...,
              [0.559447, 0.566116, 0.496751, 0.525666],
              [0.524732, 0.722651, 0.590491, 0.713421],
              [0.54931 , 0.594385, 0.516837, 0.656513]]])

    • abundance
      (chain, draw, tube, timepoint, species)
      float64
      0.1274 1.22 ... 0.9017 0.6815

      array([[[[[0.127353  , 1.22016   ],
                [0.15424   , 1.20543   ],
                [0.186704  , 1.18764   ],
                ...,
                [2.14813   , 0.112907  ],
                [2.25519   , 0.054248  ],
                [2.31199   , 0.0231206 ]],
      
               [[0.126321  , 1.2256    ],
                [0.137249  , 1.2201    ],
                [0.149114  , 1.21412   ],
                ...,
                [0.510479  , 1.03215   ],
                [0.553288  , 1.01059   ],
                [0.599493  , 0.98732   ]],
      
               [[0.136553  , 1.2528    ],
                [0.152453  , 1.24199   ],
                [0.170161  , 1.22996   ],
                ...,
      ...
                ...,
                [0.439792  , 0.982459  ],
                [0.476624  , 0.960567  ],
                [0.516343  , 0.936959  ]],
      
               [[0.139224  , 1.05019   ],
                [0.154932  , 1.04207   ],
                [0.172378  , 1.03305   ],
                ...,
                [0.808993  , 0.704025  ],
                [0.889967  , 0.662175  ],
                [0.977118  , 0.617132  ]],
      
               [[0.126328  , 1.19049   ],
                [0.140655  , 1.18108   ],
                [0.156579  , 1.17063   ],
                ...,
                [0.744468  , 0.784673  ],
                [0.820086  , 0.735029  ],
                [0.901656  , 0.681478  ]]]]])

  • Indexes: (10)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

    • ln_mu_max_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_mu_max_z_dim_0'))

    • ln_ks_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_ks_z_dim_0'))

    • ln_gamma_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_gamma_z_dim_0'))

    • tube
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], dtype='int64', name='tube'))

    • species
      PandasIndex

      PandasIndex(Index([1, 2], dtype='int64', name='species'))

    • sigma_y_dim_0
      PandasIndex

      PandasIndex(Index([0, 1], dtype='int64', name='sigma_y_dim_0'))

    • strain
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4], dtype='int64', name='strain'))

    • timepoint
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], dtype='int64', name='timepoint'))

  • Attributes: (4)

    created_at :

    2024-04-30T07:32:24.228168

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• posterior_predictive

  <xarray.Dataset> Size: 2MB
  Dimensions:      (chain: 4, draw: 300, measurement: 160)
  Coordinates:
    * chain        (chain) int64 32B 0 1 2 3
    * draw         (draw) int64 2kB 0 1 2 3 4 5 6 ... 293 294 295 296 297 298 299
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      yrep         (chain, draw, measurement) float64 2MB 0.2274 0.937 ... 0.9595
  Attributes:
      created_at:                 2024-04-30T07:32:24.234666
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • yrep
      (chain, draw, measurement)
      float64
      0.2274 0.937 ... 0.6091 0.9595

      array([[[0.227384, 0.936965, 0.436816, ..., 0.983157, 0.668725,
               0.826308],
              [0.261367, 1.15937 , 0.485356, ..., 0.927863, 0.810587,
               0.949939],
              [0.237106, 1.09835 , 0.396129, ..., 1.09271 , 0.679098,
               0.777113],
              ...,
              [0.209302, 1.18392 , 0.330663, ..., 0.818786, 0.827183,
               0.718728],
              [0.205393, 1.49241 , 0.397251, ..., 0.885399, 0.630624,
               0.85129 ],
              [0.209828, 1.01091 , 0.368422, ..., 0.794482, 0.651933,
               0.791165]],
      
             [[0.212651, 1.31444 , 0.372666, ..., 0.95774 , 0.686714,
               0.985028],
              [0.232429, 0.992496, 0.461549, ..., 0.965484, 0.846621,
               0.868622],
              [0.217889, 1.33868 , 0.410758, ..., 0.745689, 0.726656,
               0.646724],
      ...
              [0.201336, 1.15256 , 0.372924, ..., 1.1165  , 0.582028,
               0.906945],
              [0.219615, 1.09883 , 0.392869, ..., 0.835978, 0.641543,
               0.798561],
              [0.219139, 0.948146, 0.391069, ..., 0.902791, 0.734089,
               0.859884]],
      
             [[0.226279, 1.02197 , 0.413365, ..., 0.930944, 0.760439,
               0.790015],
              [0.214464, 1.36287 , 0.409723, ..., 0.960035, 0.747794,
               0.935883],
              [0.230627, 1.20867 , 0.391851, ..., 0.877454, 0.735588,
               0.731156],
              ...,
              [0.244746, 1.26408 , 0.43012 , ..., 0.797974, 0.715362,
               0.756669],
              [0.210013, 1.252   , 0.368124, ..., 0.842344, 0.768577,
               0.746415],
              [0.251364, 1.30447 , 0.44961 , ..., 1.05276 , 0.609121,
               0.959459]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-30T07:32:24.234666

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• log_likelihood

  <xarray.Dataset> Size: 2MB
  Dimensions:      (chain: 4, draw: 300, measurement: 160)
  Coordinates:
    * chain        (chain) int64 32B 0 1 2 3
    * draw         (draw) int64 2kB 0 1 2 3 4 5 6 ... 293 294 295 296 297 298 299
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      llik         (chain, draw, measurement) float64 2MB 3.07 0.4682 ... 1.037
  Attributes:
      created_at:                 2024-04-30T07:32:24.235885
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • llik
      (chain, draw, measurement)
      float64
      3.07 0.4682 2.422 ... 1.184 1.037

      array([[[ 3.06974  ,  0.468154 ,  2.42202  , ..., -0.52844  ,
                1.86399  ,  0.785454 ],
              [ 2.89394  ,  0.836349 ,  2.18481  , ...,  0.280494 ,
                1.75281  ,  0.486793 ],
              [ 3.10592  ,  0.605888 ,  2.53602  , ...,  0.844739 ,
                1.86992  ,  0.201968 ],
              ...,
              [ 2.94527  ,  0.326955 ,  2.57049  , ..., -0.570148 ,
                1.55771  ,  1.01998  ],
              [ 2.90857  ,  0.0470462,  2.53981  , ...,  0.119252 ,
                1.75207  ,  0.569022 ],
              [ 2.81414  ,  0.129617 ,  2.50832  , ...,  0.361471 ,
                1.35924  ,  0.608006 ]],
      
             [[ 3.18796  ,  0.727392 ,  2.54549  , ..., -0.341253 ,
                1.40833  ,  1.14795  ],
              [ 3.012    ,  0.206913 ,  2.47588  , ..., -0.0408692,
                1.73821  ,  0.792903 ],
              [ 2.9483   , -0.128337 ,  2.23154  , ...,  1.16819  ,
                1.89705  , -1.08372  ],
      ...
              [ 2.97165  ,  0.512323 ,  2.70612  , ..., -0.579588 ,
                0.778223 ,  0.943887 ],
              [ 2.89355  ,  0.0899549,  2.28445  , ..., -0.0749024,
                1.69561  ,  0.725514 ],
              [ 2.71393  , -0.192312 ,  2.41595  , ...,  0.315336 ,
                1.70167  ,  0.350371 ]],
      
             [[ 2.8408   ,  0.418232 ,  2.41493  , ..., -0.16898  ,
                1.25455  ,  1.03437  ],
              [ 3.01917  ,  0.336305 ,  2.30847  , ..., -0.505104 ,
                1.8999   ,  0.996751 ],
              [ 3.05213  ,  1.03249  ,  2.19536  , ...,  1.16087  ,
                1.85772  , -0.841761 ],
              ...,
              [ 2.98202  ,  0.844719 ,  2.31255  , ...,  0.603364 ,
                1.32581  ,  0.496724 ],
              [ 3.00547  ,  0.236982 ,  2.62469  , ..., -0.193662 ,
                1.87816  ,  0.37846  ],
              [ 2.77859  ,  0.276103 ,  1.99299  , ..., -0.413784 ,
                1.18399  ,  1.03708  ]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-30T07:32:24.235885

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• sample_stats

  <xarray.Dataset> Size: 61kB
  Dimensions:          (chain: 4, draw: 300)
  Coordinates:
    * chain            (chain) int64 32B 0 1 2 3
    * draw             (draw) int64 2kB 0 1 2 3 4 5 6 ... 294 295 296 297 298 299
  Data variables:
      lp               (chain, draw) float64 10kB 98.7 100.3 98.41 ... 107.8 99.97
      acceptance_rate  (chain, draw) float64 10kB 0.8923 0.9493 ... 0.9548 0.6988
      step_size        (chain, draw) float64 10kB 0.02326 0.02326 ... 0.02116
      tree_depth       (chain, draw) int64 10kB 8 8 7 7 7 8 7 7 ... 8 8 7 7 7 8 8
      n_steps          (chain, draw) int64 10kB 255 255 127 255 ... 127 255 255
      diverging        (chain, draw) bool 1kB False False False ... False False
      energy           (chain, draw) float64 10kB -70.99 -70.01 ... -83.65 -75.14
  Attributes:
      created_at:                 2024-04-30T07:32:24.232711
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

  • Data variables: (7)
    • lp
      (chain, draw)
      float64
      98.7 100.3 98.41 ... 107.8 99.97

      array([[ 98.6994, 100.282 ,  98.4108, ...,  94.3663,  95.6063,  98.9217],
             [101.716 , 101.835 ,  99.2573, ..., 113.606 , 104.797 , 108.535 ],
             [104.881 , 116.195 , 104.    , ..., 103.933 ,  91.8939,  98.9594],
             [ 99.8044,  93.0295,  91.9602, ..., 102.789 , 107.759 ,  99.965 ]])

    • acceptance_rate
      (chain, draw)
      float64
      0.8923 0.9493 ... 0.9548 0.6988

      array([[0.892305, 0.949287, 0.988558, ..., 0.872621, 0.998322, 0.544854],
             [0.998353, 0.923934, 0.998974, ..., 0.962084, 0.995258, 0.990927],
             [0.993958, 0.990483, 0.987299, ..., 0.959857, 0.888056, 0.871114],
             [0.997017, 0.998715, 0.970947, ..., 0.995319, 0.954803, 0.698783]])

    • step_size
      (chain, draw)
      float64
      0.02326 0.02326 ... 0.02116 0.02116

      array([[0.0232577, 0.0232577, 0.0232577, ..., 0.0232577, 0.0232577,
              0.0232577],
             [0.026324 , 0.026324 , 0.026324 , ..., 0.026324 , 0.026324 ,
              0.026324 ],
             [0.0234513, 0.0234513, 0.0234513, ..., 0.0234513, 0.0234513,
              0.0234513],
             [0.0211602, 0.0211602, 0.0211602, ..., 0.0211602, 0.0211602,
              0.0211602]])

    • tree_depth
      (chain, draw)
      int64
      8 8 7 7 7 8 7 7 ... 8 8 8 7 7 7 8 8

      array([[8, 8, 7, ..., 7, 7, 7],
             [7, 7, 7, ..., 7, 7, 7],
             [8, 7, 7, ..., 7, 7, 7],
             [7, 7, 8, ..., 7, 8, 8]])

    • n_steps
      (chain, draw)
      int64
      255 255 127 255 ... 255 127 255 255

      array([[255, 255, 127, ..., 127, 127, 127],
             [127, 127, 127, ..., 255, 127, 255],
             [255, 127, 127, ..., 127, 127, 127],
             [255, 255, 255, ..., 127, 255, 255]])

    • diverging
      (chain, draw)
      bool
      False False False ... False False

      array([[False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False]])

    • energy
      (chain, draw)
      float64
      -70.99 -70.01 ... -83.65 -75.14

      array([[-70.9876, -70.0079, -74.0583, ..., -64.4323, -66.379 , -68.7524],
             [-71.7556, -75.488 , -72.4687, ..., -88.9214, -84.9861, -86.2475],
             [-82.8876, -85.5851, -86.2919, ..., -77.9567, -78.2329, -66.4122],
             [-78.5389, -65.0723, -65.1918, ..., -74.4399, -83.6473, -75.1433]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

  • Attributes: (4)

    created_at :

    2024-04-30T07:32:24.232711

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  

Diagnostics: is the posterior ok?

First check the sample_stats group to see if there were any divergent transitions and if the lp parameter converged.

az.summary(idata_posterior.sample_stats)

/home/nicholas/courses/bayesian_statistics_for_computational_biology/.venv/lib/python3.12/site-packages/arviz/stats/diagnostics.py:592: RuntimeWarning: invalid value encountered in scalar divide
  (between_chain_variance / within_chain_variance + num_samples - 1) / (num_samples)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
lp	102.940	5.559	92.235	112.809	0.246	0.174	516.0	819.0	1.000000e+00
acceptance_rate	0.937	0.087	0.775	1.000	0.002	0.002	1226.0	1063.0	1.000000e+00
step_size	0.024	0.002	0.021	0.026	0.001	0.001	4.0	4.0	5.859337e+15
tree_depth	7.300	0.458	7.000	8.000	0.118	0.085	15.0	15.0	1.200000e+00
n_steps	186.200	64.273	127.000	255.000	15.662	11.272	17.0	17.0	1.170000e+00
diverging	0.000	0.000	0.000	0.000	0.000	0.000	1200.0	1200.0	NaN
energy	-76.800	7.424	-89.448	-62.775	0.349	0.247	451.0	754.0	1.000000e+00

Next check the parameter-by-parameter summary

az.summary(idata_posterior)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
ln_mu_max_z[0]	1.977	0.460	1.155	2.877	0.019	0.013	602.0	819.0	1.00
ln_mu_max_z[1]	-1.284	0.482	-2.114	-0.336	0.020	0.014	560.0	719.0	1.01
ln_mu_max_z[2]	-0.091	0.473	-0.966	0.857	0.018	0.013	655.0	800.0	1.00
ln_mu_max_z[3]	-0.026	0.466	-0.836	0.812	0.021	0.015	518.0	736.0	1.01
ln_ks_z[0]	-0.157	0.748	-1.705	1.112	0.022	0.020	1113.0	916.0	1.00
...	...	...	...	...	...	...	...	...	...
abundance[16, 18, 2]	0.746	0.046	0.662	0.834	0.001	0.001	1549.0	1064.0	1.00
abundance[16, 19, 1]	0.877	0.039	0.804	0.953	0.001	0.001	1313.0	1050.0	1.00
abundance[16, 19, 2]	0.695	0.049	0.610	0.790	0.001	0.001	1521.0	1064.0	1.00
abundance[16, 20, 1]	0.968	0.045	0.885	1.058	0.001	0.001	1237.0	1046.0	1.00
abundance[16, 20, 2]	0.639	0.052	0.548	0.739	0.001	0.001	1483.0	1095.0	1.00

704 rows × 9 columns

Show posterior intervals

prior_abundances = idata_posterior.posterior["abundance"]

n_sample = 20
chains = rng.choice(prior_abundances.coords["chain"].values, n_sample)
draws = rng.choice(prior_abundances.coords["draw"].values, n_sample)
f, axes = plot_sim(true_abundance, fake_measurements, species_to_ax)

for ax, species_i in zip(axes, species):
    for tube_j in tubes:
        for chain, draw in zip(chains, draws):
            timeseries = prior_abundances.sel(chain=chain, draw=draw, tube=tube_j, species=species_i)
            ax.plot(
                timepoints.values, 
                timeseries.values,
                alpha=0.5, color="skyblue", zorder=-1
            )
f.savefig("img/monod_posteriors.png")

look at the posterior

The next few cells use arviz’s plot_posterior function to plot the marginal posterior distributions for some of the model’s parameters:

f, axes = plt.subplots(1, 4, figsize=[10, 4])
axes = az.plot_posterior(
    idata_posterior,
    kind="hist",
    bins=20,
    var_names=["gamma"],
    ax=axes,
    point_estimate=None,
    hdi_prob="hide"
)
for ax, true_value in zip(axes, true_param_values["gamma"]):
    ax.axvline(true_value, color="red")

f, axes = plt.subplots(1, 4, figsize=[10, 4])
axes = az.plot_posterior(
    idata_posterior,
    kind="hist",
    bins=20,
    var_names=["mu_max"],
    ax=axes,
    point_estimate=None,
    hdi_prob="hide"
)
for ax, true_value in zip(axes, true_param_values["mu_max"]):
    ax.axvline(true_value, color="red")

f, axes = plt.subplots(1, 4, figsize=[10, 4])
axes = az.plot_posterior(
    idata_posterior,
    kind="hist",
    bins=20,
    var_names=["ks"],
    ax=axes,
    point_estimate=None,
    hdi_prob="hide"
)
for ax, true_value in zip(axes, true_param_values["ks"]):
    ax.axvline(true_value, color="red")

References

Allen, Rosalind J, and Bartłomiej Waclaw. 2019. “Bacterial Growth: A Statistical Physicist’s Guide.” Reports on Progress in Physics. Physical Society (Great Britain) 82 (1): 016601. https://doi.org/10.1088/1361-6633/aae546.

Timonen, Juho, Nikolas Siccha, Ben Bales, Harri Lähdesmäki, and Aki Vehtari. 2022. “An Importance Sampling Approach for Reliable and Efficient Inference in Bayesian Ordinary Differential Equation Models.” arXiv. https://doi.org/10.48550/arXiv.2205.09059.